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Fh(i)2592 4211 y Fk(v)2626 4221 y Fh(j)2659 4211 y Fj(\))2742 4179 y Fn(.)49 b(This)32 b(ratio)j(is)e(minimized)150 4302 y(when)h(the)h(angle)h(b)s(et)m(w)m(een)g Fm(v)1195 4316 y Fk(i)1258 4302 y Fn(and)e Fm(v)1483 4316 y Fk(j)1555 4302 y Fn(is)h Fm(\022)g Fl(')e Fn(2)p Fm(:)p Fn(33)j(radians,)g (giving)g(a)g(ratio)g(of)f Fm(\013)e Fl(')g Fn(0)p Fm(:)p Fn(87856.)58 b(By)150 4415 y(linearit)m(y)32 b(of)e(exp)s(ectation,)i (the)e(exp)s(ected)h(n)m(um)m(b)s(er)d(of)i(edges)h(cut)f(b)m(y)g(the)g (random)f(h)m(yp)s(erplane)g(is)h(at)150 4528 y(least)i Fm(\013)e Fn(times)g(the)h(v)-5 b(alue)31 b(of)f(geometric)i(program,)d (giving)i(an)g Fm(\013)f Fn(appro)m(ximation)h(for)f(MAX)h(CUT.)291 4641 y(W)-8 b(e)46 b(remark)d(that)j(a)f(random)d(h)m(yp)s(erplane)i (can)h(b)s(e)f(c)m(hosen)h(algorithmically)i(b)m(y)d(c)m(ho)s(osing)i (a)150 4754 y(random)24 b(unit)h(v)m(ector)i Fm(r)s Fn(,)f(whic)m(h)f (implicitly)h(de\014nes)e(the)i(h)m(yp)s(erplane)e Fl(f)p Fm(x)p Fl(j)p Fm(xr)29 b Fn(=)c(0)p Fl(g)p Fn(.)40 b(See)25 b(details)i(in)e([8].)1927 5247 y(2)p eop %%Page: 3 3 3 2 bop 150 -58 a Fo(2)137 b(Researc)l(h)47 b(questions)150 145 y Fn(In)30 b(general,)h(our)f(researc)m(h)h(goal)h(is)e(the)h (follo)m(wing.)291 258 y Fg(Researc)m(h)36 b(goal:)k Fn(Impro)m(v)m(e)30 b(the)g(appro)m(ximation)g(ratio)i(of)e(MAX)h(CUT)f (b)s(ey)m(ond)g Fm(\013)c Fl(')f Fn(0)p Fm(:)p Fn(87856.)291 371 y(As)35 b(an)h(in)m(termediate)g(step,)h(w)m(e)f(wish)e(to)i (obtain)g(a)g(b)s(etter)f(understanding)f(of)i(algorithm)f(GW.)150 484 y(Let)c(us)f(note)h(that)g(there)f(are)h(four)f(v)-5 b(alues)30 b(that)h(in)m(terest)h(us:)261 664 y(1.)46 b(The)30 b(v)-5 b(alue)31 b(of)g(the)f(SDP)-8 b(,)31 b(denoted)f(b)m(y)h(sdp)n(.)261 843 y(2.)46 b(The)30 b(v)-5 b(alue)31 b(of)g(the)f(true)g(maxim)m(um)e(cut,)i(denoted)h(b)m (y)f(opt.)261 1022 y(3.)46 b(The)39 b(exp)s(ected)g(v)-5 b(alue)40 b(of)f(the)g(cut)h(pro)s(duced)d(b)m(y)i(the)g(randomized)f (algorithm,)k(denoted)d(b)m(y)377 1134 y(exp.)261 1313 y(4.)46 b(The)30 b(actual)i(v)-5 b(alue)31 b(of)f(the)h(cut)f(pro)s (duced)f(b)m(y)h(the)h(algorithm,)g(denoted)f(b)m(y)g(alg)q(.)291 1494 y(W)-8 b(e)35 b(note)f(that)g(w)m(e)g(ma)m(y)g(assume)e(that)i (for)f(the)h(GW)g(algorithm,)h(alg)d Fl(\025)e Fn(exp)j(\(p)s(erhaps)g (up)f(to)j(a)150 1607 y(negligibly)d(small)d(additiv)m(e)j(term\).)40 b(There)30 b(are)h(t)m(w)m(o)g(w)m(a)m(ys)h(of)e(establishing)h(suc)m (h)f(a)h(result.)261 1788 y(1.)46 b Fg(Indep)s(enden)m(t)26 b(rep)s(etitions)p Fn(.)36 b(Rep)s(eat)24 b(the)f(random)e(h)m(yp)s (erplane)h(rounding)f(tec)m(hnique)j(p)s(oly-)377 1901 y(nomially)32 b(man)m(y)f(times)h(indep)s(enden)m(tly)-8 b(,)32 b(and)f(tak)m(e)j(the)e(b)s(est)f(of)h(these)h(cuts.)45 b(As)32 b(the)g(n)m(um)m(b)s(er)377 2014 y(of)f(edges)g(in)f(a)h(cut)f (is)h(b)s(ounded)d(b)m(y)j(at)g(most)f Fm(n)2041 1981 y Fj(2)2080 2014 y Fn(,)g(the)h(v)-5 b(ariance)31 b(of)g(the)f(random)f (v)-5 b(ariable)31 b(that)377 2126 y(denotes)j(the)f(n)m(um)m(b)s(er)e (of)j(edges)g(that)g(w)m(e)f(get)i(in)e(a)g(particular)h(cut)f(is)h (also)g(b)s(ounded.)47 b(Hence)377 2239 y(with)29 b(high)g(probabilit)m (y)-8 b(,)31 b(at)f(least)h(one)e(of)h(the)f(random)f(cuts)h(has)g(v)-5 b(alue)30 b(not)f(m)m(uc)m(h)g(lo)m(w)m(er)h(than)377 2352 y(exp.)261 2531 y(2.)46 b Fg(Derandomization)p Fn(.)71 b(Using)41 b(the)g(metho)s(d)f(of)h(conditional)h(exp)s(ectations)g (one)f(can)g(deter-)377 2644 y(ministically)36 b(select)i(a)e(h)m(yp)s (erplane)e(that)j(giv)m(es)g(a)f(cut)f(with)h(v)-5 b(alue)36 b(at)g(least)h(that)f(of)g(exp,)h(up)377 2757 y(to)k(p)s(erhaps)e(a)h (negligibly)h(small)f(additiv)m(e)h(term)e(that)i(is)f(a)h(result)f(of) g(p)s(erforming)e(n)m(umeric)377 2870 y(computations)30 b(with)h(\014nite)f(precision)g([13)r(].)291 3050 y(W)-8 b(e)32 b(can)e(summarize)f(the)h(relations)i(b)s(et)m(w)m(een)f(the)f (four)g(v)-5 b(alues)31 b(as)f(follo)m(ws:)1494 3276 y(exp)25 b Fl(\024)g Fn(alg)h Fl(\024)f Fn(opt)h Fl(\024)f Fn(sdp)o Fm(:)291 3430 y Fn(The)30 b(questions)g(that)h(w)m(e)g(shall)g (study)e(are)i(the)g(follo)m(wing.)286 3610 y Fl(\017)46 b Fn(Ho)m(w)31 b(close)h(is)f(the)f(v)-5 b(alue)31 b(of)g(the)g(SDP)f (to)h(the)g(v)-5 b(alue)31 b(of)f(the)h(maxim)m(um)c(cut?)41 b(The)30 b(w)m(orst)h(case)377 3723 y(ratio)26 b(\(tak)m(en)h(o)m(v)m (er)f(all)g(p)s(ossible)e(graphs\))h(of)g(opt)p Fm(=)p Fn(sdp)g(will)g(b)s(e)f(called)i(the)f Ff(inte)-5 b(gr)g(ality)30 b(r)-5 b(atio)p Fn(.)40 b(An)377 3836 y(in)m(tegralit)m(y)32 b(ratio)d(close)h(to)f(1)g(means)e(that)i(w)m(e)g(can)g(estimate)g(the) g(v)-5 b(alue)29 b(of)g(the)f(maxim)m(um)e(cut)377 3949 y(within)i(a)g(small)f(error)h(term,)f(without)h(necessarily)h (exhibiting)f(a)g(cut)g(that)h(is)f(nearly)g(optimal.)286 4128 y Fl(\017)46 b Fn(Ho)m(w)38 b(close)h(is)e(the)h(v)-5 b(alue)38 b(of)f(the)h(cut)g(found)e(b)m(y)h(the)h(GW)g(algorithm)f (\(after)h(the)g(rounding\))377 4241 y(to)45 b(the)f(v)-5 b(alue)45 b(of)f(the)g(maxim)m(um)d(cut?)81 b(The)44 b(w)m(orst)g(case)h(ratio)g(of)g(alg)q Fm(=)p Fn(opt)f(is)h(called)g (the)377 4354 y Ff(appr)-5 b(oximation)37 b(r)-5 b(atio)p Fn(.)291 4534 y(The)28 b(analysis)h(of)g(Go)s(emans)f(and)g(Williamson) h(sho)m(ws)f(that)i(the)e(exp)s(ected)h(n)m(um)m(b)s(er)e(of)i(edges)g (cut)150 4647 y(b)m(y)j(the)h(random)d(h)m(yp)s(erplane)i(rounding)f (tec)m(hnique)i(is)f(at)i(least)f(an)f Fm(\013)h Fn(fraction)g(of)g (the)f(v)-5 b(alue)33 b(of)g(the)150 4760 y(semide\014nite)d (relaxation.)42 b(That)30 b(is,)h(they)f(sho)m(w)h(that)g(for)f(ev)m (ery)h(graph)1778 4890 y(exp)p 1778 4930 139 4 v 1779 5013 a(sdp)1952 4951 y Fl(\025)25 b Fm(\013:)1927 5247 y Fn(3)p eop %%Page: 4 4 4 3 bop 150 -58 a Fn(As)30 b(there)g(exists)g(at)h(least)g(one)f(cut)f (ac)m(hieving)j(this)d(exp)s(ectation,)j(the)e(in)m(tegralit)m(y)j (ratio)d(of)g(the)g(SDP)150 55 y(is)36 b(no)f(w)m(orse)h(than)f Fm(\013)g Fl(')f Fn(0)p Fm(:)p Fn(87856.)59 b(The)35 b(w)m(orst)h(in)m(tegralit)m(y)i(ratio)f(kno)m(wn)e(for)g(the)h(SDP)g (is)f(roughly)150 168 y(0)p Fm(:)p Fn(8845.)41 b(It)23 b(is)h(obtained)g(for)f(the)h(5-cycle.)40 b(Asso)s(ciate)25 b(with)e(eac)m(h)i(v)m(ertex)f(of)g(the)g(5-cycle)h(a)f(unit)f(v)m (ector,)150 281 y(and)32 b(la)m(y)h(all)g(these)g(v)m(ectors)h(as)e(a)h (symmetric)d(star)j(in)f(the)g(plane,)h(in)f(the)h(\(cyclic\))h(order)e (1)p Fm(;)15 b Fn(3)p Fm(;)g Fn(5)p Fm(;)g Fn(2)p Fm(;)g Fn(4.)150 394 y(An)m(y)39 b(t)m(w)m(o)g(v)m(ectors)h(that)f(corresp)s (ond)f(to)h(adjacen)m(t)h(v)m(ertices)g(mak)m(e)e(an)h(angle)g(of)g (360)27 b Fl(\001)f Fn(2)p Fm(=)p Fn(5)40 b(=)e(144,)150 506 y(with)25 b(cosine)h Fl(\000)p Fn(0)p Fm(:)p Fn(8090)i(and)d(con)m (tribution)h(0)p Fm(:)p Fn(9045)i(to)e(SDP)-8 b(.)26 b(Hence)g(the)g(v)-5 b(alue)26 b(of)g(the)f(SDP)h(is)f(roughly)150 619 y(5)e Fl(\001)g Fn(0)p Fm(:)p Fn(9045)34 b(=)d(4)p Fm(:)p Fn(5225,)38 b(whereas)c(the)h(maxim)m(um)c(cut)j(cuts)g(4)h (edges,)h(giving)f(an)f(in)m(tegralit)m(y)j(ratio)e(of)150 732 y(roughly)30 b(0)p Fm(:)p Fn(8845.)43 b(In)30 b(this)g(man)m (uscript)f(w)m(e)i(sho)m(w:)150 944 y Fg(Theorem)k(1)46 b Ff(F)-7 b(or)33 b(every)f Fm(\017)25 b(>)g Fn(0)p Ff(,)32 b(ther)-5 b(e)33 b(is)f(a)h(gr)-5 b(aph)34 b(for)e(which)h(the)g(inte) -5 b(gr)g(ality)34 b(r)-5 b(atio)33 b(of)g(the)f(SDP)g(of)150 1057 y(Go)-5 b(emans)35 b(and)e(Wil)5 b(liamson)34 b(is)e(no)i(b)-5 b(etter)33 b(than)g Fm(\013)21 b Fn(+)f Fm(\017)p Ff(.)291 1268 y Fn(W)-8 b(e)38 b(no)m(w)f(turn)f(to)i(the)g(appro)m(ximation)f (ratio)h(for)f(the)g(GW)h(algorithm.)60 b(Recall)39 b(that)f(w)m(e)f (ma)m(y)150 1381 y(assume)c(that)i(alg)e Fl(\025)e Fn(exp.)52 b(Using)35 b(also)g(the)g(fact)g(that)g(opt)d Fl(\024)f Fn(sdp)o(,)k(the)g(analysis)g(of)f(Go)s(emans)g(and)150 1494 y(Williamson)26 b(implies)f(that)i(the)e(appro)m(ximation)h(ratio) h(is)f(at)g(least)h Fm(\013)p Fn(.)40 b(But)26 b(is)g(it)g(b)s(etter)g (than)f Fm(\013)p Fn(?)40 b(This)150 1607 y(question)31 b(w)m(as)h(studied)e(b)m(y)h(Karlo\013)g([12)q(].)43 b(F)-8 b(or)32 b(ev)m(ery)g Fm(\017)26 b(>)g Fn(0,)32 b(he)e(constructed)i(a)f(graph)f(that)i(has)f(the)150 1720 y(follo)m(wing)h(prop)s(erties:)261 1932 y(1.)46 b(sdp)24 b(=)h(opt)q(.)261 2119 y(2.)46 b(The)30 b(graph)g(has)g(an)g (optimal)h(em)m(b)s(edding)d(on)j(the)f(sphere)g(for)g(whic)m(h)g(exp) 25 b Fl(\024)g Fn(\()p Fm(\013)c Fn(+)f Fm(\017)p Fn(\)sdp)o(.)291 2331 y(Th)m(us)40 b(Karlo\013)i(w)m(as)f(able)h(to)g(sho)m(w)f(that)h (the)f(ratio)i(exp)o Fm(=)p Fn(opt)f(can)g(b)s(e)e(arbitrarily)i(close) g(to)g Fm(\013)p Fn(.)150 2444 y(Ho)m(w)m(ev)m(er,)g(this)c(do)s(es)g (not)h(sho)m(w)f(the)g(appro)m(ximation)g(ratio)h(alg)q Fm(=)p Fn(opt)g(is)g(as)f(bad)f(as)i Fm(\013)p Fn(,)i(b)s(ecause)d(of) 150 2556 y(the)e(p)s(ossibilit)m(y)g(that)g(for)g(these)g(graphs)f(alg) g Fm(>)f Fn(exp.)56 b(Indeed,)37 b(it)f(w)m(as)g(observ)m(ed)g(that)g (in)f(Karlo\013)7 b('s)150 2669 y(graphs)31 b(there)g(are)h(h)m(yp)s (erplanes)e(that)i(pro)s(duce)e(the)i(optimal)f(cut,)h(and)f(the)h(p)s (ossibilit)m(y)g(that)f(either)150 2782 y(indep)s(enden)m(t)25 b(rep)s(etitions)i(or)f(derandomization)g(will)g(pro)s(duce)f(suc)m(h)h (a)h(cut)f(has)g(not)h(b)s(een)e(ruled)g(out.)291 2895 y(In)33 b(this)g(man)m(uscript)g(w)m(e)g(sho)m(w)h(ho)m(w)f(Karlo\013)7 b('s)35 b(examples)e(can)h(b)s(e)e(extended)i(to)g(sho)m(w)g(that)g (the)150 3008 y(appro)m(ximation)c(ratio)i(is)e(indeed)g(arbitrarily)h (close)g(to)g Fm(\013)p Fn(.)150 3220 y Fg(Theorem)k(2)46 b Ff(F)-7 b(or)35 b(every)g Fm(\017)29 b(>)f Fn(0)p Ff(,)36 b(ther)-5 b(e)35 b(is)g(a)g(gr)-5 b(aph)37 b(and)e(an)g(optimal)i(emb) -5 b(e)g(dding)36 b(of)f(this)g(gr)-5 b(aph)36 b(on)150 3333 y(the)d(spher)-5 b(e)34 b(such)f(that)h(for)f Fn(ev)m(ery)g Ff(hyp)-5 b(erplane,)35 b(alg)26 b Fl(\024)f Fn(\()p Fm(\013)c Fn(+)f Fm(\017)p Fn(\))p Ff(opt)q(.)291 3544 y Fn(Exp)s(erience)g(sho)m(ws)h(that)g(for)f(linear)h(programming)d (relaxations)k(and)e(semide\014nite)f(programming)150 3657 y(relaxations,)35 b(one)e(do)s(es)g(not)g(exp)s(ect)g(to)h(\014nd) d(a)i(rounding)e(tec)m(hnique)j(that)f(giv)m(es)h(a)g(b)s(etter)e (appro)m(xi-)150 3770 y(mation)27 b(ratio)i(than)f(the)f(in)m(tegralit) m(y)k(ratio.)41 b(Hence)28 b(w)m(e)g(are)g(lead)h(to)f(seeking)h(w)m(a) m(ys)f(of)g(impro)m(ving)f(the)150 3883 y(in)m(tegralit)m(y)g(ratio)f (of)f(the)g(SDP)f(of)h(GW.)g(This)f(can)h(b)s(e)f(done)g(b)m(y)h (adding)f(to)i(the)e(semide\014nite)g(program)150 3996 y(constrain)m(ts)i(that)g(are)f(v)-5 b(alid)26 b(for)f(ev)m(ery)h(true) e(cut.)40 b(Some)24 b(sp)s(eci\014c)h(constrain)m(ts,)i(the)e(so)h (called)g Ff(triangle)150 4109 y(c)-5 b(onstr)g(aints)p Fn(,)30 b(w)m(ere)d(suggested)g(and)f(analysed)g(in)g([6)q(],)h(where)f (it)h(w)m(as)g(sho)m(wn)f(that)g(they)h(lead)g(to)g(b)s(etter)150 4222 y(appro)m(ximation)34 b(algorithms)h(for)f(problems)f(related)i (to)h(MAX)f(CUT,)f(suc)m(h)g(as)h(MAX)g(DICUT)g(and)150 4335 y(MAX)f(2SA)-8 b(T.)34 b(Ho)m(w)m(ev)m(er,)j(it)e(is)e(not)h(kno)m (wn)f(whether)h(the)f(triangle)i(constrain)m(ts)g(help)e(in)h(impro)m (ving)150 4448 y(the)f(in)m(tegralit)m(y)j(ratio)e(for)e(MAX)i(CUT)e(b) s(ey)m(ond)g Fm(\013)p Fn(.)49 b(F)-8 b(or)34 b(the)f(5-cycle,)j(the)d (in)m(tegralit)m(y)j(ratio)e(of)f(the)150 4560 y(SDP)39 b(with)g(triangle)i(constrain)m(ts)f(is)78 b(1.)68 b(W)-8 b(e)41 b(sho)m(w)e(that)h(if)f(the)g(triangle)i(constrain)m(ts)f(do)f (indeed)150 4673 y(impro)m(v)m(e)30 b(the)h(in)m(tegralit)m(y)i(ratio)e (for)f(MAX)h(CUT,)g(this)f(impro)m(v)m(emen)m(t)f(is)i(limited.)150 4885 y Fg(Theorem)k(3)46 b Ff(F)-7 b(or)37 b(the)g(SDP)f(for)h(MAX)f (CUT)g(with)h(triangle)h(c)-5 b(onstr)g(aints,)40 b(ther)-5 b(e)37 b(ar)-5 b(e)37 b(gr)-5 b(aphs)39 b(with)150 4998 y(inte)-5 b(gr)g(ality)34 b(r)-5 b(atio)35 b(no)e(b)-5 b(etter)33 b(than)g(r)-5 b(oughly)35 b Fn(0)p Fm(:)p Fn(891)p Ff(.)1927 5247 y Fn(4)p eop %%Page: 5 5 5 4 bop 291 -58 a Fn(As)43 b(for)f(the)g(ratio)i(exp)p Fm(=)p Fn(opt)f(\(with)g(the)f(random)f(h)m(yp)s(erplane)g(rounding)h (tec)m(hnique\),)k(Karlo\013)150 55 y(sho)m(ws)27 b(that)g(it)h (remains)e(arbitrarily)h(close)h(to)g Fm(\013)f Fn(ev)m(en)h(when)e (arbitrary)h(v)-5 b(alid)27 b(constrain)m(ts)h(are)g(added)150 168 y(to)h(the)f(SDP)-8 b(.)29 b(Our)e(Theorem)g(2)h(that)h(sho)m(ws)f (alg)q Fm(=)p Fn(opt)e Fl(')f Fm(\013)k Fn(also)g(extends)f(to)h(the)f (case)h(that)g(arbitrary)150 281 y(v)-5 b(alid)31 b(constrain)m(ts)g (are)g(added)f(\(ev)m(en)h(when)e(the)i(b)s(est)f(h)m(yp)s(erplane)f (is)i(used\).)150 524 y Fb(2.1)112 b(Organization)37 b(of)g(this)g(pap)s(er)150 696 y Fn(In)27 b(Section)h(3)g(w)m(e)g(pro)m (v)m(e)g(that)g(the)g(in)m(tegralit)m(y)i(ratio)f(of)e(the)h(GW)g(SDP)f (is)h Fm(\013)p Fn(,)h(and)d(that)j(with)e(triangle)150 809 y(constrain)m(ts)43 b(the)f(in)m(tegralit)m(y)j(ratio)e(is)f(no)g (b)s(etter)g(than)f(roughly)h(0)p Fm(:)p Fn(891.)77 b(In)42 b(Section)g(4)h(w)m(e)f(sho)m(w)150 921 y(that)35 b(the)f(appro)m (ximation)g(ratio)i(is)e(no)g(b)s(etter)g(than)g Fm(\013)p Fn(.)53 b(Section)35 b(4)g(is)f(comp)s(osed)f(of)h(sev)m(eral)i(parts.) 150 1034 y(In)d(Section)i(4.1)g(w)m(e)g(review)f(the)g(triangle)i (constrain)m(ts)f(and)e(the)i(consequences)f(of)g(adding)g(them)f(to) 150 1147 y(the)28 b(SDP)-8 b(.)29 b(In)f(Section)h(4.2)g(w)m(e)g (describ)s(e)f(other)g(v)-5 b(alid)29 b(constrain)m(ts,)h(and)d(metho)s (ds)g(of)h(obtaining)h(SDP)150 1260 y(solutions)c(that)g(satisfy)g (them.)38 b(In)24 b(Section)i(4.3)f(w)m(e)g(pro)s(duce)f(examples)g (similar)g(in)h(spirit)f(to)h(Karlo\013)7 b('s)150 1373 y(examples.)39 b(The)24 b(pro)s(ofs)h(in)f(this)h(section)i(are)e (elemen)m(tary)h(and)f(do)g(not)g(in)m(v)m(olv)m(e)j(the)d(dual)g(of)g (the)h(SDP)-8 b(,)150 1486 y(nor)26 b(the)i(notion)f(of)g(eigen)m(v)-5 b(alues.)42 b(In)26 b(Section)i(4.4)g(w)m(e)f(sho)m(w)g(ho)m(w)g(these) g(and)g(Karlo\013)7 b('s)27 b(examples)g(can)150 1599 y(b)s(e)i(extended)g(to)i(pro)m(v)m(e)f(Theorem)e(2.)41 b(This)29 b(extension)h(in)m(v)m(olv)m(es)h(graphs)e(that)h(are)g(not)g (connected,)h(a)150 1712 y(tec)m(hnicalit)m(y)i(that)e(is)g(dealt)g (with)f(in)g(Section)h(4.5.)150 1955 y Fb(2.2)112 b(Notation)36 b(and)j(con)m(v)m(en)m(tions)150 2127 y Fn(W)-8 b(e)33 b(shall)g(use)f(the)g(con)m(v)m(en)m(tion)j(\(unless)d(explicitly)i (stated)f(otherwise\))g(that)g(the)f(distance)h(b)s(et)m(w)m(een)150 2240 y(t)m(w)m(o)41 b(v)m(ectors)g(is)e(the)h(angle)g(b)s(et)m(w)m(een) g(them.)67 b(Lik)m(ewise,)43 b(the)c(distance)h(b)s(et)m(w)m(een)g(t)m (w)m(o)h(p)s(oin)m(ts)e(on)h(a)150 2353 y(sphere)30 b(is)g(the)h(angle) g(they)f(mak)m(e)h(as)f(view)m(ed)h(from)e(the)i(cen)m(ter)g(of)g(the)f (sphere.)291 2466 y(W)-8 b(e)45 b(shall)e(often)h(not)g(care)g(ab)s (out)f(precise)h(calculations,)49 b(and)43 b(ma)m(y)g(neglect)i(terms)e (that)h(are)150 2579 y(insigni\014can)m(t)25 b(to)g(our)e(main)g (results.)39 b(Hence)25 b(w)m(e)f(shall)h(sometimes)e(use)h(notation)h (suc)m(h)f(as)h Fl(')p Fn(,)g(or)f(terms)150 2691 y(suc)m(h)34 b(as)h(\\roughly",)i(to)f(denote)f(the)g(fact)h(that)f(the)g(results)g (presen)m(ted)f(are)h(not)g(strictly)h(accurate,)150 2804 y(but)d(are)i(close)h(enough)e(to)g(the)h(true)f(results)g(for)g (our)f(purp)s(oses.)51 b(Lik)m(ewise,)36 b(w)m(e)f(use)f(the)g(term)f Ff(high)150 2917 y(pr)-5 b(ob)g(ability)25 b Fn(in)e(the)h(sense)f(of)g (\\probabilit)m(y)h(high)f(enough)g(to)g(mak)m(e)h(the)f(rest)g(of)g (the)h(argumen)m(ts)e(w)m(ork",)150 3030 y(without)30 b(exactly)i(quan)m(tifying)e(it.)41 b(In)30 b(most)f(cases)i(\(but)e(p) s(erhaps)g(not)h(all\),)i(this)d(means)g(probabilit)m(y)150 3143 y(that)g(can)g(b)s(e)e(made)h(arbitrarily)h(close)g(to)g(1)g(b)m (y)g(taking)g(appropriate)f(v)-5 b(alues)29 b(\(large)h(enough)e(or)g (small)150 3256 y(enough,)i(as)h(the)g(case)g(ma)m(y)f(b)s(e\))g(for)g (the)h(parameters)e(in)m(v)m(olv)m(ed.)150 3542 y Fo(3)137 b(The)46 b(In)l(tegralit)l(y)h(Ratio)150 3745 y Fn(In)33 b(this)g(section)i(w)m(e)f(sho)m(w)f(that)i(the)e(in)m(tegralit)m(y)k (ratio)d(opt)p Fm(=)p Fn(sdp)f(ma)m(y)g(b)s(e)g(arbitrarily)h(close)h (to)f Fm(\013)d Fl(')150 3858 y Fn(0)p Fm(:)p Fn(87856.)40 b(F)-8 b(or)21 b(ev)m(ery)h Fm(\017)j(>)g Fn(0,)e(w)m(e)d(construct)h (a)g(graph)f(\(that)h(dep)s(ends)e(on)h Fm(\017)p Fn(\))g(for)g(whic)m (h)g(opt)q Fm(=)p Fn(sdp)k Fl(\024)h Fm(\013)p Fn(+)p Fm(\017)p Fn(.)150 3971 y(Unlik)m(e)38 b(the)g(5-cycle)h(example)e(for) g(whic)m(h)g(the)g(in)m(tegralit)m(y)j(ratio)f(w)m(as)e(computed)f (exactly)-8 b(,)42 b(w)m(e)c(will)150 4084 y(not)32 b(b)s(e)g(able)g (to)h(exactly)h(compute)d(opt)h(and)g(sdp)n(.)46 b(Instead,)32 b(w)m(e)h(shall)f(b)s(ound)e(sdp)h(from)f(b)s(elo)m(w)j(and)150 4197 y(opt)e(from)e(ab)s(o)m(v)m(e,)i(th)m(us)f(b)s(ounding)f(the)h(in) m(tegralit)m(y)k(ratio)d(from)e(ab)s(o)m(v)m(e.)291 4310 y(Fix)24 b(an)f(arbitrary)f Fm(\017)j(>)g Fn(0.)39 b(W)-8 b(e)24 b(no)m(w)f(explain)g(ho)m(w)g(to)h(construct)g(a)f(graph)f Fm(G)h Fn(satisfying)g(opt)q Fm(=)p Fn(sdp)h Fl(\024)150 4423 y Fm(\013)19 b Fn(+)g Fm(\017)p Fn(.)40 b(F)-8 b(or)31 b(our)e(graph)g(w)m(e)h(will)g(need)g(a)g(lo)m(w)m(er)h(b)s(ound)c(on)j (sdp)o(.)40 b(This)29 b(will)h(b)s(e)f(done)h(b)m(y)f(exhibiting)i(a) 150 4536 y(feasible)c(\(though)f(not)g(necessarily)h(optimal\))f (solution)h(to)g(the)f(geometric)h(program)e(for)h(MAX)g(CUT.)150 4649 y(Rather)35 b(than)g(\014rst)f(construct)i(a)f(graph)f Fm(G)h Fn(and)f(then)h(try)g(to)g(\014nd)f(go)s(o)s(d)h(feasible)h (solutions)f(to)h(the)150 4762 y(SDP)-8 b(,)31 b(w)m(e)g(shall)f(ha)m (v)m(e)i(a)f(geometric)g(solution)g(in)f(mind,)f(and)g(then)h(build)g Fm(G)g Fn(around)f(it.)291 4874 y(Recall)c(that)f Fm(\022)j Fl(')e Fn(2)p Fm(:)p Fn(33)g(radians)e(is)h(the)f(w)m(orst)h(angle)g (for)f(the)h(GW)g(algorithm.)38 b(That)24 b(is,)h(if)e(the)g(unit)150 4987 y(v)m(ectors)34 b(asso)s(ciated)h(with)e(the)g(endp)s(oin)m(ts)f (of)h(an)g(edge)h(mak)m(e)f(an)f(angle)i(of)g Fm(\022)s Fn(,)f(then)f(the)h(probabilit)m(y)1927 5247 y(5)p eop %%Page: 6 6 6 5 bop 150 -58 a Fm(\022)s(=\031)30 b Fn(that)d(a)h(random)d(h)m(yp)s (erplane)h(cuts)h(the)g(edge)h(is)f Fm(\013)g Fn(times)g(the)g(con)m (tribution)h(\(1)13 b Fl(\000)g Fn(cos)k Fm(\022)s Fn(\))p Fm(=)p Fn(2)28 b(of)f(the)150 55 y(edge)k(to)g(the)g(v)-5 b(alue)31 b(of)f(the)h(SDP)-8 b(.)1802 144 y(2)p Fm(\022)p 1615 184 465 4 v 1615 268 a Fn(\(1)21 b Fl(\000)f Fn(cos)c Fm(\022)s Fn(\))p Fm(\031)2115 205 y Fn(=)25 b Fm(\013:)150 421 y Fn(Let)31 b Fm(\022)356 435 y Fj(1)420 421 y Fm(<)25 b(\022)j(<)d(\022)726 435 y Fj(2)795 421 y Fn(b)s(e)30 b(suc)m(h)g(that)h(the)f(in)m(terv)-5 b(al)32 b([)p Fm(\022)1874 435 y Fj(1)1913 421 y Fm(;)15 b(\022)1996 435 y Fj(2)2036 421 y Fn(])30 b(is)h(the)f(largest)i(satisfying)1729 613 y(2)p Fm(\022)1820 580 y Fi(0)p 1542 654 488 4 v 1542 737 a Fn(\(1)21 b Fl(\000)f Fn(cos)c Fm(\022)1917 711 y Fi(0)1940 737 y Fn(\))p Fm(\031)2065 675 y Fl(\024)25 b Fm(\013)c Fn(+)f Fm(\017)1266 b Fn(\(1\))150 929 y(for)30 b(ev)m(ery)h Fm(\022)572 943 y Fj(1)637 929 y Fl(\024)25 b Fm(\022)779 896 y Fi(0)826 929 y Fl(\024)g Fm(\022)965 943 y Fj(2)1004 929 y Fn(.)291 1042 y(Consider)k(a)h(unit)g(sphere)f Fm(S)1281 1009 y Fk(d)p Fi(\000)p Fj(1)1441 1042 y Fn(in)h Fm(R)1617 1009 y Fk(d)1657 1042 y Fn(.)41 b(Distribute)30 b Fm(n)f Fn(p)s(oin)m(ts)h(uniformly)e(on)i(the)g(sphere,)f(where)150 1155 y Fm(n)45 b(>>)f(d)p Fn(.)77 b(When)42 b Fm(d)j Fn(=)g(2)e(\(namely)-8 b(,)46 b(the)c(sphere)g(is)g(a)h(circle\),)k (the)42 b Fm(n)g Fn(p)s(oin)m(ts)g(can)h(b)s(e)e(placed)i(at)150 1268 y(in)m(terv)-5 b(als)34 b(of)g(size)g(2)p Fm(\031)s(=n)p Fn(.)50 b(When)33 b Fm(d)d(>)g Fn(2,)35 b(they)e(can)h(b)s(e)e (distributed)h(uniformly)e(at)j(random.)48 b(The)32 b Fm(n)150 1381 y Fn(p)s(oin)m(ts)27 b(represen)m(t)f(the)h Fm(n)f Fn(v)m(ertices)j(of)e Fm(G)p Fn(,)g(and)f(their)h(placemen)m(t)g (on)g(the)f(sphere)g(represen)m(ts)h(a)g(feasible)150 1494 y(solution)32 b(to)h(the)f(geometric)h(program.)44 b(No)m(w)33 b(join)f(t)m(w)m(o)h(v)m(ertices)g(b)m(y)f(an)g(edge)g(if)g (the)g(angle)h(that)g(the)150 1607 y(corresp)s(onding)f(unit)h(v)m (ectors)h(mak)m(es)f(is)g(b)s(et)m(w)m(een)h Fm(\022)2010 1621 y Fj(1)2082 1607 y Fn(and)e Fm(\022)2304 1621 y Fj(2)2343 1607 y Fn(.)49 b(This)32 b(completes)h(the)h(description)f (of)150 1720 y(the)e(graph)e Fm(G)p Fn(,)h(and)g(the)h(description)f (of)h(a)f(feasible)i(solution)f(for)f(the)g(geometric)i(program.)291 1833 y(Equation)23 b(\(1\))h(implies)e(that)i(for)f(the)g(graph)f Fm(G)g Fn(and)h(its)g(geometric)h(em)m(b)s(edding,)f(exp)p Fm(=)p Fn(sdp)h Fl(\024)h Fm(\013)6 b Fn(+)f Fm(\017)p Fn(.)150 1945 y(Moreo)m(v)m(er,)26 b(if)21 b(the)h(p)s(oin)m(ts)f(of)h Fm(G)e Fn(are)i(dense)f(and)g(uniform)e(on)j(the)f(sphere,)i(all)f(h)m (yp)s(erplanes)f(cut)g(roughly)150 2058 y(the)27 b(same)e(n)m(um)m(b)s (er)f(of)j(edges,)h(and)e(hence)g(the)h(n)m(um)m(b)s(er)d(of)i(edges)h (cut)g(b)m(y)f(an)m(y)h(h)m(yp)s(erplane)e(is)h(at)i(most)150 2171 y(a)34 b(fraction)g(of)g(\()p Fm(\013)23 b Fn(+)f Fm(\017)p Fn(\))33 b(of)h(the)f(v)-5 b(alue)34 b(of)g(the)g(SDP)-8 b(.)33 b(This)g(last)h(statemen)m(t)h(is)e(true)g(regardless)h(of)g (the)150 2284 y(question)f(of)g(whether)f(the)h(feasible)h(solution)f (that)h(w)m(e)f(exhibited)g(for)f(the)h(SDP)g(is)g(optimal)g(or)f(not.) 150 2397 y(Hence,)38 b(w)m(e)e(w)m(ould)g(b)s(e)f(done)g(if)h(w)m(e)g (could)g(sho)m(w)f(that)h(for)g Fm(G)p Fn(,)h(the)e(maxim)m(um)e(cut)j (is)g(indeed)f(a)h(cut)150 2510 y(obtained)31 b(b)m(y)g(a)g(h)m(yp)s (erplane.)41 b(Sho)m(wing)31 b(this)g(is)f(the)h(crux)g(of)g(the)g(pro) s(of.)41 b(There)30 b(are)i(certain)f(pitfalls)150 2623 y(that)g(should)e(b)s(e)h(a)m(v)m(oided,)i(and)e(they)g(are)h(describ)s (ed)f(next.)291 2736 y(Consider)40 b Fm(d)p Fn(,)j(the)d(dimension)f (of)i(the)f(em)m(b)s(edding.)69 b(The)40 b(simplest)f(c)m(hoice)j(is)f (to)g(tak)m(e)h Fm(d)g Fn(=)f(2.)150 2849 y(Unfortunately)-8 b(,)37 b(this)d(do)s(es)h(not)g(w)m(ork.)54 b(Observ)m(e)35 b(that)g Fm(\022)g Fl(')d Fn(2)p Fm(:)p Fn(33)k(is)f(v)m(ery)g(close)i (to)e(3)p Fm(\031)s(=)p Fn(4.)55 b(If)35 b Fm(\022)i Fn(w)m(as)150 2962 y(exactly)32 b(3)p Fm(\031)s(=)p Fn(4)g(and)f(w)m(e) g(w)m(ould)f(tak)m(e)i Fm(\022)1502 2976 y Fj(1)1567 2962 y Fn(=)26 b Fm(\022)1707 2976 y Fj(2)1771 2962 y Fn(=)g Fm(\022)33 b Fn(and)d Fm(n)g Fn(divisible)h(b)m(y)f(8,)i(then)e (the)h(graph)f Fm(G)g Fn(w)m(ould)150 3075 y(just)f(decomp)s(ose)g(in)m (to)i(a)f(collection)i(of)e(8-cycles,)i(and)d(hence)h(the)g(maxim)m(um) d(cut)i(w)m(ould)h(include)f Ff(al)5 b(l)150 3187 y Fn(edges.)41 b(As)28 b Fm(\022)f Fl(6)p Fn(=)e(3)p Fm(\031)s(=)p Fn(4,)31 b Fm(G)c Fn(is)i(not)f(a)h(bipartite)g(graph,)f(but)g(is)g(v)m(ery)h (close)h(to)f(it.)40 b(By)29 b(cutting)g(the)g(cycle)150 3300 y(in)m(to)e(8)g(equal)g(segmen)m(ts,)g(eac)m(h)h(of)e(size)h Fm(\031)s(=)p Fn(4,)h(and)e(placing)h(the)g(v)m(ertices)h(of)e(ev)m(en) h(n)m(um)m(b)s(ered)d(segmen)m(ts)150 3413 y(on)29 b(one)g(side)f(of)h (the)g(cut)g(and)f(those)h(of)g(o)s(dd)f(n)m(um)m(b)s(ered)e(segmen)m (ts)j(on)g(the)g(other)g(side,)g(w)m(e)g(cut)g(m)m(uc)m(h)150 3526 y(more)k(than)g(a)h Fm(\022)s(=\031)i Fn(fraction)e(of)f(the)h (edges.)51 b(Hence)34 b(cuts)f(generated)i(b)m(y)e(h)m(yp)s(erplanes)f (are)i(far)f(from)150 3639 y(optimal)d(for)g(this)h(graph.)40 b(\(This)30 b(observ)-5 b(ation)31 b(w)m(as)g(apparen)m(tly)f(\014rst)g (made)g(b)m(y)g(Go)s(emans.\))291 3752 y(T)-8 b(o)38 b(o)m(v)m(ercome)g(this)f(problem,)h(w)m(e)g(m)m(ust)e(tak)m(e)j Fm(d)e(>)f Fn(2.)61 b(In)37 b(our)g(pro)s(of,)h Fm(d)f Fn(is)h(tak)m(en)g(to)g(b)s(e)f(some)150 3865 y(function)28 b(that)h(gro)m(ws)h(with)e(1)p Fm(=\017)p Fn(.)41 b(It)28 b(is)h(an)g(in)m(teresting)h(op)s(en)d(question)i(whether)f(some)g (\014xed)g Fm(d)h Fn(\(e.g.,)150 3978 y Fm(d)d Fn(=)f(3?\))41 b(su\016ces)30 b(for)g(all)h(v)-5 b(alues)31 b(of)f Fm(\017)p Fn(.)291 4091 y(Another)37 b(issue)f(of)h(imp)s(ortance)f(is)h(the)g (distribution)f(of)h(p)s(oin)m(ts)f(on)h(the)g(sphere.)58 b(Graphs)36 b(that)150 4204 y(are)j(comp)s(osed)f(of)g(the)h Fm(n)g Fn(=)g(2)1261 4171 y Fk(d)1340 4204 y Fn(p)s(oin)m(ts)g (represen)m(ted)g(as)g(the)g(v)m(ectors)2756 4168 y Fj(1)p 2726 4183 96 4 v 2726 4192 a Fi(p)p 2784 4192 37 3 v 2784 4249 a Fk(d)2831 4204 y Fl(f\006)p Fn(1)p Fl(g)3037 4171 y Fk(d)3117 4204 y Fn(w)m(ere)g(considered)150 4332 y(in)30 b([12)q(,)h(1].)41 b(Corollary)31 b(15)g(in)g(Section)g(4.2)g (sho)m(ws)f(that)h(these)g(so)g(called)g(\\h)m(yp)s(ercub)s(es)e(em)m (b)s(eddings")150 4445 y(cannot)c(b)s(e)g(used)e(to)j(demonstrate)e(an) h(in)m(tegralit)m(y)j(ratio)d(other)g(than)g(1.)39 b(\(This)24 b(fact)i(w)m(as)f(kno)m(wn)g(prior)150 4558 y(to)k(this)f(pap)s(er.\)) 39 b(F)-8 b(or)29 b(our)e(construction)i(the)f(n)m(um)m(b)s(er)e(of)i (p)s(oin)m(ts)g(will)g(also)h(b)s(e)f(exp)s(onen)m(tial)h(in)f Fm(d)p Fn(,)h(but)150 4671 y(they)i(will)f(b)s(e)g(distributed)f(at)j (random)c(on)i(the)h(sphere.)1927 5247 y(6)p eop %%Page: 7 7 7 6 bop 150 -58 a Fb(3.1)112 b(A)37 b(Dense)h(V)-9 b(ersion)150 114 y Fn(In)32 b(this)g(section)i(w)m(e)e(sho)m(w)h(a)g(construction)g (with)f(a)h(large)g(n)m(um)m(b)s(er)d(of)j(p)s(oin)m(ts)f(\(more)g (than)g(a)h(simple)150 226 y(exp)s(onen)m(tial)39 b(in)f(the)h (dimension)d Fm(d)p Fn(\).)65 b(Later,)41 b(in)d(Section)h(3.2)g(w)m(e) g(will)g(sho)m(w)f(a)g(construction)h(with)150 339 y(few)m(er)30 b(p)s(oin)m(ts.)40 b(The)28 b(latter)j(construction)f(is)f(not)h (needed)f(in)g(order)f(to)i(analyse)g(the)g(in)m(tegralit)m(y)i(ratio) 150 452 y(of)i(the)h(SDP)f(used)g(in)g(the)g(GW)h(algorithm,)h(but)d (will)i(b)s(e)f(used)f(in)h(analysing)h(the)g(in)m(tegralit)m(y)i (ratio)150 565 y(for)30 b(the)h(SDP)f(with)g(triangle)h(constrain)m (ts.)291 678 y(T)-8 b(o)30 b(simplify)f(notation)i(and)e(not)h(deal)g (with)f(fractions)h(of)g Fm(\017)p Fn(,)g(w)m(e)g(shall)g(sho)m(w)g(an) f(in)m(tegralit)m(y)k(ratio)150 791 y(of)e Fm(\013)20 b Fn(+)g Fm(O)s Fn(\()p Fm(\017)p Fn(\),)31 b(rather)f(than)h Fm(\013)20 b Fn(+)g Fm(\017)p Fn(.)291 904 y(W)-8 b(e)41 b(measure)d(distances)i(b)s(et)m(w)m(een)g(p)s(oin)m(ts)f(on)g(the)h (unit)f(sphere)g(b)m(y)g(the)g(angle)i(b)s(et)m(w)m(een)f(their)150 1017 y(resp)s(ectiv)m(e)31 b(unit)f(v)m(ectors.)291 1130 y(The)g(parameters)g(of)g(our)g(construction)h(are)g(c)m(hosen)g(as)f (follo)m(ws.)286 1317 y Fl(\017)46 b Fn(The)f(angles)h Fm(\022)910 1331 y Fj(1)995 1317 y Fn(and)e Fm(\022)1229 1331 y Fj(2)1314 1317 y Fn(are)h(c)m(hosen)h(so)g(as)f(to)h(satisfy)g (Equation)g(\(1\).)86 b(\(Using)46 b(a)g(T)-8 b(a)m(ylor)377 1430 y(expansion)30 b(of)h(\(1\))g(around)f Fm(\022)i Fn(sho)m(ws)e(that)h(taking)h(them)d(in)h(the)h(range)f Fm(\022)23 b Fl(\006)d Fm(O)s Fn(\()3178 1365 y Fl(p)p 3254 1365 37 4 v 65 x Fm(\017)p Fn(\))30 b(su\016ces.\))286 1618 y Fl(\017)46 b Fn(The)31 b(dimension)f Fm(d)i Fn(is)g(c)m(hosen)g (as)g(the)f(smallest)h(suc)m(h)f(that)h(in)g Fm(S)2654 1585 y Fk(d)p Fi(\000)p Fj(1)2816 1618 y Fn(the)f(\()p Fm(d)22 b Fl(\000)f Fn(1\)-dimensional)377 1731 y(v)m(olume)36 b(of)g(a)g(cap)g(of)g(radius)f(\()p Fm(\031)28 b Fl(\000)23 b Fm(\022)1698 1745 y Fj(2)1737 1731 y Fn(\))36 b(is)g(an)g Fm(\017)p Fn(-fraction)h(of)f(the)g(v)m(olume)f(of)h(a)h(cap)f(of)g (radius)377 1844 y(\()p Fm(\031)25 b Fl(\000)20 b Fm(\022)623 1858 y Fj(1)662 1844 y Fn(\).)44 b(\(F)-8 b(rom)31 b(the)g(v)-5 b(alues)32 b(of)f Fm(\022)1617 1858 y Fj(1)1688 1844 y Fn(and)f Fm(\022)1908 1858 y Fj(2)1979 1844 y Fn(and)g(from)g(Lemma)g (9)h(it)h(will)g(follo)m(w)g(that)g(taking)377 1957 y Fm(d)26 b Fl(')546 1884 y Fe(p)p 629 1884 128 4 v 73 x Fn(1)p Fm(=\017)16 b Fn(log)r(\(1)p Fm(=\017)p Fn(\))32 b(su\016ces.\))286 2144 y Fl(\017)46 b Fn(W)-8 b(e)32 b(also)f(use)f(a)h(small)f(enough)g(parameter)g Fm(\015)35 b Fn(\(e.g.,)e Fm(\015)d(<)25 b(\017=d)p Fn(\))31 b(satisfying:)480 2332 y Fg({)45 b Fn(The)27 b(v)m(olume)g(of)h(a)f(cap)h(of)f(radius)g Fm(\031)17 b Fl(\000)d Fm(\022)1976 2346 y Fj(1)2029 2332 y Fl(\000)g Fm(\015)32 b Fn(is)27 b(at)h(least)h(a)f(1)14 b Fl(\000)g Fm(\017)27 b Fn(fraction)h(of)g(the)f(v)m(olume)577 2445 y(of)k(a)g(cap)f(of)h(radius)e Fm(\031)24 b Fl(\000)19 b Fm(\022)1509 2459 y Fj(1)1569 2445 y Fn(+)h Fm(\015)5 b Fn(.)480 2591 y Fg({)45 b Fn(F)-8 b(or)31 b(ev)m(ery)h Fm(\022)1027 2558 y Fi(0)1049 2591 y Fn(,)f(cos\()p Fm(\022)1307 2558 y Fi(0)1330 2591 y Fn(\))g(and)f(cos\()p Fm(\022)1775 2558 y Fi(0)1818 2591 y Fl(\006)20 b Fm(\015)5 b Fn(\))31 b(di\013er)f(b)m(y)g(an)h(additiv)m(e)g(factor)h(of)e(at)h(most)f Fl(\006)p Fm(\017)p Fn(.)291 2779 y(W)-8 b(e)32 b(are)e(no)m(w)h(ready) f(to)h(describ)s(e)f(the)h(graph)e Fm(G)h Fn(for)g(whic)m(h)g(opt)q Fm(=)p Fn(sdp)24 b Fl(\024)h Fm(\013)c Fn(+)f Fm(O)s Fn(\()p Fm(\017)p Fn(\).)261 2991 y(1.)46 b(Cho)s(ose)30 b(the)h(parameters)f Fm(\022)1364 3005 y Fj(1)1403 2991 y Fn(,)g Fm(\022)1501 3005 y Fj(2)1540 2991 y Fn(,)h Fm(d)g Fn(and)e Fm(\015)36 b Fn(as)30 b(describ)s(ed)g(ab)s(o)m(v)m(e.) 261 3179 y(2.)46 b(P)m(artition)37 b Fm(S)831 3146 y Fk(d)p Fi(\000)p Fj(1)996 3179 y Fn(in)m(to)f Fm(n)c Fn(=)g(\()1420 3134 y Fk(O)r Fj(\(1\))p 1420 3158 146 4 v 1473 3210 a Fk(\015)1576 3179 y Fn(\))1611 3146 y Fk(d)1687 3179 y Fn(equal)j(v)m(olume)g(cells,)i(where)d(eac)m(h)i (cell)g(has)f(diameter)f(at)377 3292 y(most)c Fm(\015)5 b Fn(.)41 b(\(See)31 b(Lemma)d(21)j(in)g(the)f(app)s(endix)f(for)h (more)g(details.\))261 3479 y(3.)46 b(In)30 b(eac)m(h)h(cell)h(pic)m(k) f(one)g(arbitrary)f(p)s(oin)m(t)g(to)h(represen)m(t)g(a)f(v)m(ertex)i (of)e Fm(G)p Fn(.)261 3667 y(4.)46 b(Edges)31 b(connect)g(those)g (pairs)f(of)g(v)m(ertices)i(whose)f(distance)g(is)f(b)s(et)m(w)m(een)h Fm(\022)2976 3681 y Fj(1)3045 3667 y Fn(and)f Fm(\022)3265 3681 y Fj(2)3304 3667 y Fn(.)150 3879 y Fg(Theorem)35 b(4)46 b Ff(F)-7 b(or)31 b(the)f(gr)-5 b(aph)32 b Fm(G)d Ff(describ)-5 b(e)g(d)32 b(ab)-5 b(ove,)31 b(the)f(inte)-5 b(gr)g(ality)32 b(r)-5 b(atio)31 b(opt)q Fm(=)p Ff(sdp)h(of)e(the)h(GW) f(SDP)150 3992 y(is)j(at)g(most)g Fm(\013)21 b Fn(+)f Fm(O)s Fn(\()p Fm(\017)p Fn(\))p Ff(.)291 4205 y Fn(T)-8 b(o)36 b(pro)m(v)m(e)h(Theorem)d(4)i(w)m(e)g(\014rst)f(consider)h Fm(G)1915 4219 y Fk(c)1949 4205 y Fn(,)h(a)f(con)m(tin)m(uous)h (analogue)g(of)f(the)g(graph)f Fm(G)p Fn(.)56 b(W)-8 b(e)150 4318 y(ma)m(y)31 b(think)f(of)i Fm(G)765 4332 y Fk(c)830 4318 y Fn(as)g(follo)m(ws.)44 b(The)30 b(v)m(ertices)j(of)f Fm(G)1973 4332 y Fk(c)2038 4318 y Fn(are)f(all)h(p)s(oin)m(ts)g(of)f Fm(S)2757 4285 y Fk(d)p Fi(\000)p Fj(1)2887 4318 y Fn(.)44 b(The)30 b(edges)i(of)f Fm(G)3563 4332 y Fk(c)3629 4318 y Fn(are)150 4431 y(pairs)e(of)g(v)m(ertices)h(of)f(distance)h(at)g (least)g Fm(\022)1620 4445 y Fj(1)1659 4431 y Fn(.)40 b(\(A)m(t)30 b(the)g(momen)m(t)d(w)m(e)j(ignore)f Fm(\022)2841 4445 y Fj(2)2880 4431 y Fn(.)40 b(It)29 b(will)h(b)s(e)e(considered)150 4544 y(again)34 b(later.\))52 b(Because)34 b(the)g(graph)f(has)g(an)g (in\014nite)h(\(and)f(uncoun)m(table\))h(n)m(um)m(b)s(er)d(of)i(v)m (ertices)j(and)150 4656 y(edges,)e(w)m(e)e(shall)h(not)f(use)g(terms)g (suc)m(h)g(as)g(the)h(\\n)m(um)m(b)s(er")d(of)j(edges,)g(but)f(rather)g (the)h(\\measure")f(of)150 4769 y(edges.)1927 5247 y(7)p eop %%Page: 8 8 8 7 bop 291 -58 a Fn(Denote)28 b(b)m(y)f Fm(\026)f Fn(the)g(normalized) g(\()p Fm(d)12 b Fl(\000)g Fn(1\)-dimensional)28 b(natural)f(measure)e (on)i Fm(S)3051 -91 y Fk(d)p Fi(\000)p Fj(1)3208 -58 y Fn(and)f(b)m(y)g Fm(\026)3558 -91 y Fj(2)3624 -58 y Fn(the)150 55 y(induced)34 b(pro)s(duct)f(measure)h Fm(\026)22 b Fl(\002)h Fm(\026)35 b Fn(on)f Fm(S)1649 22 y Fk(d)p Fi(\000)p Fj(1)1803 55 y Fl(\002)23 b Fm(S)1958 22 y Fk(d)p Fi(\000)p Fj(1)2088 55 y Fn(.)54 b(Giv)m(en)35 b(a)g(\(measurable\))g(subset)f Fm(A)h Fn(of)f Fm(S)3619 22 y Fk(d)p Fi(\000)p Fj(1)150 168 y Fn(and)c(a)h Fm(\032)f Fn(b)s(et)m(w)m(een)h(0)g(and)e Fm(\031)s Fn(,)i(w)m(e)g(also)g(denote) 361 372 y Fm(\026)416 386 y Fk(\032)456 372 y Fn(\()p Fm(A)p Fn(\))26 b(=)f Fm(\026)771 334 y Fj(2)810 372 y Fn(\()p Fl(f)p Fn(\()p Fm(x;)15 b(y)s Fn(\);)48 b Fm(x)25 b Fl(2)g Fm(A;)45 b(y)29 b Fl(62)24 b Fm(A;)46 b Fn(the)31 b(distance)g(b)s(et)m(w)m(een)g Fm(x)f Fn(and)g Fm(y)j Fn(is)d(at)i(least)f Fm(\032)p Fl(g)p Fn(\))p Fm(:)291 576 y Fn(F)-8 b(or)37 b Fm(G)532 590 y Fk(c)566 576 y Fn(,)g(w)m(e)f(w)m(an)m(t)h(to)f(partition)h(it)f(in)m(to)h Fm(A)e Fn(and)2092 553 y(\026)2068 576 y Fm(A)g Fn(while)h(maximizing)f (the)h(measure)e(of)i(edges)150 689 y(cut.)41 b(This)29 b(corresp)s(onds)g(to)j(\014nding)d Fm(A)h Fn(that)h(maximizes)f Fm(\026)2256 703 y Fk(\032)2296 689 y Fn(\()p Fm(A)p Fn(\).)150 902 y Fg(Theorem)35 b(5)46 b Ff(Fix)37 b(an)h Fm(a)f Ff(b)-5 b(etwe)g(en)38 b(0)g(and)g(1)g(and)g(a)f Fm(\032)h Ff(b)-5 b(etwe)g(en)37 b(0)h(and)g Fm(\031)s Ff(.)56 b(Then)38 b(the)f(maximum)i(of)150 1015 y Fm(\026)205 1029 y Fk(\032)245 1015 y Fn(\()p Fm(A)p Fn(\))g Ff(wher)-5 b(e)40 b Fm(A)e Ff(r)-5 b(anges)40 b(over)e(al)5 b(l)39 b(\(me)-5 b(asur)g(able\))41 b(subsets)d(of)h Fm(S)2454 982 y Fk(d)p Fi(\000)p Fj(1)2623 1015 y Ff(of)g(me)-5 b(asur)g(e)39 b Fm(a)g Ff(is)f(attaine)-5 b(d)40 b(for)150 1127 y(a\(ny\))34 b(c)-5 b(ap)33 b(of)g(me)-5 b(asur)g(e)34 b Fm(a)p Ff(.)291 1340 y Fn(Theorem)h(5)i(can)f(b)s(e)g(deduce)g(from)e (results)i(of)h(Baernstein)g(and)e(T)-8 b(a)m(ylor)38 b([3)q(].)58 b(W)-8 b(e)37 b(indicate)h(the)150 1453 y(pro)s(of)28 b(in)g(the)h(app)s(endix.)39 b(It)29 b(is)g(v)m(ery)g(m)m (uc)m(h)f(motiv)-5 b(ated)29 b(b)m(y)f(a)h(pro)s(of)f(of)h(the)g(isop)s (erimetric)g(inequalit)m(y)150 1566 y(on)h(the)h(sphere)e(\(see)j(for)e (example)g([4)q(])g(or)h([17)q(]\).)150 1778 y Fg(Corollary)i(6)46 b Ff(Fix)33 b(a)h Fm(\032)f Ff(b)-5 b(etwe)g(en)34 b(0)g(and)g Fm(\031)s Ff(.)44 b(Then)34 b(the)f(maximum)i(of)e Fm(\026)2725 1792 y Fk(\032)2765 1778 y Fn(\()p Fm(A)p Fn(\))i Ff(wher)-5 b(e)34 b Fm(A)f Ff(r)-5 b(anges)35 b(over)150 1891 y(al)5 b(l)35 b(\(me)-5 b(asur)g(able\))37 b(subsets)d(of)h Fm(S)1300 1858 y Fk(d)p Fi(\000)p Fj(1)1465 1891 y Ff(is)g(attaine)-5 b(d)36 b(for)g(a\(ny\))f(c)-5 b(ap)36 b(of)f(me)-5 b(asur)g(e)35 b Fn(1)p Fm(=)p Fn(2)h Ff(\(namely,)g(a)f(half-)150 2004 y(spher)-5 b(e\).)291 2217 y Fg(Pro)s(of:)37 b Fn(Theorem)21 b(5)i(implies)f(that)h(the)f(cut)h(that)f(maximizes)g(the)g(measure)g (of)g(edges)h(cut)f(is)h(a)f(cap.)150 2330 y(W)-8 b(e)35 b(sho)m(w)e(that)h(of)f(all)i(caps,)f(the)g(cap)f(that)h(cuts)g(the)f (highest)h(measure)e(of)i(edges)g(is)f(a)h(half)f(sphere.)150 2443 y(Consider)j(an)m(y)i(other)f(cap)h Fm(C)7 b Fn(.)60 b(Without)38 b(loss)f(of)h(generalit)m(y)h(w)m(e)e(ma)m(y)g(assume)f (that)i Fm(\026)p Fn(\()p Fm(C)7 b Fn(\))36 b Fm(<)g Fn(1)p Fm(=)p Fn(2,)150 2555 y(as)d(otherwise)g(w)m(e)g(consider)1184 2532 y(\026)1163 2555 y Fm(C)7 b Fn(,)33 b(the)g(complemen)m(t)e(of)i Fm(C)7 b Fn(.)47 b(Let)33 b Fm(z)j Fn(b)s(e)c(the)h(cen)m(ter)h(of)e Fm(C)7 b Fn(,)33 b(and)f(consider)150 2668 y(a)38 b(cap)f Fm(H)44 b Fn(of)38 b(measure)e(1)p Fm(=)p Fn(2)j(\(half)e(sphere\))g (cen)m(tered)h(at)g Fm(z)t Fn(.)62 b(Ev)m(ery)38 b(p)s(oin)m(t)f(in)g Fm(H)32 b Fl(n)25 b Fm(C)44 b Fn(has)37 b(a)g(higher)150 2781 y(measure)29 b(of)i(neigh)m(b)s(ors)e(in)1147 2758 y(\026)1123 2781 y Fm(H)k Fn(=)1351 2758 y(\026)1328 2781 y Fm(H)1426 2717 y Fe(T)1531 2758 y Fn(\026)1510 2781 y Fm(C)k Fn(then)30 b(in)g Fm(C)7 b Fn(.)40 b(This)29 b(implies)h(that)h(b)m(y)f(joining)g Fm(H)d Fl(n)21 b Fm(C)37 b Fn(with)30 b Fm(C)150 2894 y Fn(the)h(measure)e(of)h(edges)h (cut)g(do)s(es)f(not)h(decrease,)g(pro)m(ving)g Fm(\026)2299 2908 y Fk(\032)2339 2894 y Fn(\()p Fm(H)7 b Fn(\))26 b Fl(\025)f Fm(\026)2669 2908 y Fk(\032)2709 2894 y Fn(\()p Fm(C)7 b Fn(\).)116 b Fa(2)291 3007 y Fn(W)-8 b(e)23 b(ha)m(v)m(e)g(seen)f(that)h(for)f Fm(G)1221 3021 y Fk(c)1255 3007 y Fn(,)i(a)e(half)g(sphere)f(\(namely)-8 b(,)24 b(a)e(cap)g(pro)s(duced)f(b)m(y)g(a)i(h)m(yp)s(erplane)e(through)150 3120 y(the)33 b(cen)m(ter)g(of)g(the)f(sphere\))g(maximizes)g Fm(\026)1666 3134 y Fk(\032)1706 3120 y Fn(\()p Fm(A)p Fn(\).)48 b(Let)33 b(us)e(no)m(w)i(estimate)g(the)g(measure)e(of)i (edges)g(cut)150 3233 y(b)m(y)g(a)g(h)m(yp)s(erplane.)48 b(Note)35 b(that)e(b)m(y)g(symmetry)-8 b(,)32 b(all)i(h)m(yp)s (erplanes)e(cut)h(the)g(same)g(measure)f(of)h(edges.)150 3346 y(Hence)f(w)m(e)f(can)g(use)f(the)h(analysis)g(of)g(Go)s(emans)f (and)g(Williamson)h(regarding)g(the)g(fraction)h(of)e(edges)150 3459 y(cut)g(b)m(y)g(a)g(random)e(h)m(yp)s(erplane.)40 b(F)-8 b(or)30 b(edges)h(with)e(endp)s(oin)m(ts)g(at)i(distance)f Fm(\032)g Fn(apart,)h(this)e(fraction)i(is)150 3572 y(exactly)h Fm(\032=\031)s Fn(.)291 3685 y(Let)38 b(us)e(denote)i(b)m(y)f(opt)1149 3706 y Fk(c)1221 3685 y Fn(the)g(measure)f(of)i(edges)f(cut)h(b)m(y)f (a)h(random)d(h)m(yp)s(erplane,)j(divided)f(b)m(y)150 3797 y(the)f(total)i(measure)d(of)i(edges.)59 b(Similarly)-8 b(,)37 b(let)g(sdp)1986 3819 y Fk(c)2057 3797 y Fn(denote)g(the)f(v)-5 b(alue)37 b(of)f(the)h(SDP)e(giv)m(en)j(b)m(y)e(the)150 3910 y(iden)m(tit)m(y)c(em)m(b)s(edding)c(of)j Fm(G)1121 3924 y Fk(c)1185 3910 y Fn(on)g Fm(S)1373 3877 y Fk(d)p Fi(\000)p Fj(1)1533 3910 y Fn(divided)f(b)m(y)g(the)h(total)h(measure)d (of)i(edges.)41 b(This)30 b(v)-5 b(alue)30 b(is)h(the)150 4023 y(exp)s(ectation)36 b(of)f(\(1)23 b Fl(\000)g Fm(uv)s Fn(\))p Fm(=)p Fn(2)36 b(o)m(v)m(er)g(the)f(c)m(hoice)h(of)f(a)f (random)f(edge)i(\()p Fm(u;)15 b(v)s Fn(\).)55 b(\(W)-8 b(e)36 b(are)f(not)f(assuming)150 4136 y(here)c(that)h(this)f(is)h(the) f(optimal)h(v)-5 b(alue)31 b(for)f(the)g(SDP)g(on)h Fm(G)2244 4150 y Fk(c)2278 4136 y Fn(,)f(but)g(it)h(is)f(a)h(lo)m(w)m(er)h(b)s (ound.\))291 4249 y(Let)i(us)f(no)m(w)g(mo)s(dify)f Fm(G)1150 4263 y Fk(c)1217 4249 y Fn(b)m(y)h(remo)m(ving)h(all)g(edges)g(that)g (connect)g(pairs)f(of)h(v)m(ertices)h(of)e(distance)150 4362 y(more)e(than)g Fm(\022)635 4376 y Fj(2)674 4362 y Fn(.)44 b(Recall)33 b(that)f(the)g(dimension)e Fm(d)i Fn(is)f(c)m(hosen)h(as)g(the)g(smallest)f(suc)m(h)g(that)i(in)e Fm(S)3462 4329 y Fk(d)p Fi(\000)p Fj(1)3624 4362 y Fn(the)150 4475 y(v)m(olume)d(of)h(a)g(cap)g(of)g(radius)e(\()p Fm(\031)20 b Fl(\000)d Fm(\022)1413 4489 y Fj(2)1452 4475 y Fn(\))29 b(is)f(an)h Fm(\017)p Fn(-fraction)g(of)g(the)g(v)m (olume)f(of)h(a)g(cap)g(of)f(radius)g(\()p Fm(\031)20 b Fl(\000)c Fm(\022)3650 4489 y Fj(1)3689 4475 y Fn(\).)150 4588 y(Hence)35 b(the)f(fraction)g(of)g(edges)h(remo)m(v)m(ed)f(from)e Fm(G)1932 4602 y Fk(c)2000 4588 y Fn(is)i(at)g(most)g Fm(\017)p Fn(.)51 b(This)33 b(implies)g(that)h(neither)g(opt)3715 4610 y Fk(c)150 4701 y Fn(nor)39 b(sdp)458 4723 y Fk(c)532 4701 y Fn(c)m(hange)h(b)m(y)f(more)g(than)g Fm(\017)p Fn(.)67 b(In)39 b(particular,)j(a)e(random)d(h)m(yp)s(erplane)h(is)i (within)f(at)h(most)150 4814 y Fm(O)s Fn(\()p Fm(\017)p Fn(\))33 b(of)g(the)g(maxim)m(um)d(cut)j(for)g(the)g(new)f Fm(G)1775 4828 y Fk(c)1809 4814 y Fn(.)48 b(Observ)m(e)33 b(that)h(all)f(edges)h(no)m(w)f(mak)m(e)f(angles)i(in)f(the)150 4927 y(range)28 b([)p Fm(\022)463 4941 y Fj(1)502 4927 y Fm(;)15 b(\022)585 4941 y Fj(2)624 4927 y Fn(],)29 b(and)d(recall)j(that)f(in)f(this)g(range,)h(a)g(random)d(h)m(yp)s (erplane)h(is)i(exp)s(ected)f(to)h(cut)g(at)g(most)1927 5247 y(8)p eop %%Page: 9 9 9 8 bop 150 -58 a Fn(\()p Fm(\013)23 b Fn(+)f Fm(\017)p Fn(\)sdp)32 b(edges.)50 b(Hence)34 b(for)f(the)h(mo)s(di\014ed)d Fm(G)1909 -44 y Fk(c)1943 -58 y Fn(,)k(the)e(in)m(tegralit)m(y)j(ratio) e(is)g(at)g(most)e Fm(\013)23 b Fn(+)f Fm(O)s Fn(\()p Fm(\017)p Fn(\),)35 b(as)150 55 y(desired.)291 168 y(F)-8 b(or)30 b(simplicit)m(y)f(in)g(the)g(rest)g(of)g(the)h(presen)m (tation,)g(w)m(e)g(let)g(sdp)2501 190 y Fk(c)2565 168 y Fn(and)e(opt)2872 190 y Fk(c)2935 168 y Fn(refer)h(to)h(the)f(mo)s (di\014ed)150 281 y Fm(G)222 295 y Fk(c)294 281 y Fn(rather)38 b(than)g(the)g(original)i(one.)64 b(This)37 b(in)m(tro)s(duces)h(an)g Fm(O)s Fn(\()p Fm(\017)p Fn(\))g(error)g(in)g(the)g(analysis,)j(whic)m (h)d(is)150 394 y(negligible.)291 506 y(W)-8 b(e)32 b(no)m(w)e(return)f (to)i(our)f(graph)g Fm(G)p Fn(,)g(whic)m(h)g(is)h(a)f(discrete)i(v)m (ersion)e(of)h(the)g(mo)s(di\014ed)d Fm(G)3347 520 y Fk(c)3381 506 y Fn(.)291 619 y Fg(Pro)s(of)40 b(of)e(Theorem)h(4:)101 b Fn(Recall)35 b(that)f(to)h(obtain)f Fm(G)p Fn(,)g(w)m(e)g(partition)g Fm(S)2925 586 y Fk(d)p Fi(\000)p Fj(1)3089 619 y Fn(in)m(to)h Fm(n)e Fn(equal)h(size)150 732 y(cells,)c(where)d(eac)m(h)i(cell)g(has) e(diameter)h(at)g(most)f Fm(\015)5 b Fn(,)29 b(and)e(place)h(a)h (single)f(v)m(ertex)h(in)e(eac)m(h)i(cell.)41 b(F)-8 b(or)29 b(the)150 845 y(discrete)h(v)m(ersion)g Fm(G)p Fn(,)g(let)g(opt)1180 867 y Fk(d)1249 845 y Fn(denote)h(the)e(ratio)i (b)s(et)m(w)m(een)f(the)g(n)m(um)m(b)s(er)d(of)j(edges)g(in)f(the)h (maxim)m(um)150 958 y(cut)e(and)f(the)h(total)i(n)m(um)m(b)s(er)25 b(of)j(edges.)41 b(Let)28 b(sdp)1851 980 y Fk(d)1919 958 y Fn(denote)h(the)f(ratio)g(b)s(et)m(w)m(een)h(the)f(v)-5 b(alue)28 b(of)g(the)g(SDP)150 1071 y(giv)m(en)39 b(b)m(y)e(the)h(iden) m(tit)m(y)h(em)m(b)s(edding)c(of)j Fm(G)f Fn(on)g Fm(S)1915 1038 y Fk(d)p Fi(\000)p Fj(1)2083 1071 y Fn(and)g(the)h(total)h(n)m(um) m(b)s(er)d(of)h(edges.)63 b(W)-8 b(e)39 b(shall)150 1184 y(sho)m(w)f(that)g(sdp)724 1206 y Fk(d)802 1184 y Fl(\025)f Fn(sdp)1047 1206 y Fk(c)1107 1184 y Fl(\000)25 b Fm(\017)37 b Fn(and)h(that)g(opt)1797 1206 y Fk(d)1875 1184 y Fl(\024)g Fn(opt)2115 1206 y Fk(c)2175 1184 y Fn(+)25 b Fm(\017)p Fn(.)62 b(F)-8 b(rom)38 b(this)f(and)g(from)g(the)h(fact)g(that)150 1297 y(sdp)287 1319 y Fk(c)347 1297 y Fm(>)25 b Fn(1)p Fm(=)p Fn(2)31 b(it)f(will)g(follo)m(w)g(that)g(the)g(in)m(tegralit)m (y)i(ratio)f(in)e(the)h(discrete)g(case)h(is)e(larger)h(than)g(that)g (of)150 1410 y(the)h(con)m(tin)m(uous)g(case)g(b)m(y)f(at)h(most)f Fm(O)s Fn(\()p Fm(\017)p Fn(\),)h(pro)m(ving)g(Theorem)e(4.)291 1523 y(T)-8 b(o)31 b(analyse)g(the)g(relation)g(b)s(et)m(w)m(een)g(sdp) 1724 1544 y Fk(d)1795 1523 y Fn(and)f(sdp)2109 1544 y Fk(c)2143 1523 y Fn(,)h(partition)g(pairs)f(of)g(cells)i(in)m(to)f (\014v)m(e)g(t)m(yp)s(es:)261 1696 y(1.)46 b Ff(Ne)-5 b(ar)30 b(p)-5 b(airs.)40 b Fn(These)27 b(are)f(pairs)g(of)h(cells)h (suc)m(h)e(that)h(ev)m(ery)g(p)s(oin)m(t)f(in)g(one)h(cell)h(is)e (within)g(distance)377 1809 y(less)31 b(than)e Fm(\022)799 1823 y Fj(1)868 1809 y Fn(from)g(ev)m(ery)i(p)s(oin)m(t)f(in)f(the)i (other)f(cell.)42 b(These)30 b(pairs)f(con)m(tribute)i(0)f(b)s(oth)g (to)g(sdp)3715 1830 y Fk(c)377 1921 y Fn(and)g(to)h(sdp)802 1943 y Fk(d)842 1921 y Fn(.)261 2098 y(2.)46 b Ff(Distant)e(p)-5 b(airs.)76 b Fn(These)41 b(are)h(pairs)g(of)g(cells)g(suc)m(h)g(that)g (ev)m(ery)h(p)s(oin)m(t)e(in)h(one)g(cell)h(is)e(within)377 2211 y(distance)d(more)e(than)h Fm(\022)1234 2225 y Fj(2)1310 2211 y Fn(from)f(ev)m(ery)i(p)s(oin)m(t)f(in)g(the)g(other)h(cell.)62 b(These)37 b(pairs)g(con)m(tribute)h(0)377 2324 y(b)s(oth)30 b(to)h(sdp)840 2345 y Fk(c)905 2324 y Fn(and)f(to)h(sdp)1330 2345 y Fk(d)1370 2324 y Fn(.)261 2500 y(3.)46 b Ff(Contributing)32 b(p)-5 b(airs.)42 b Fn(These)29 b(are)g(pairs)g(of)h(cells)g(suc)m(h)f (that)g(ev)m(ery)h(p)s(oin)m(t)g(in)e(one)i(cell)g(is)f(within)377 2613 y(distance)35 b(b)s(et)m(w)m(een)h Fm(\022)1134 2627 y Fj(1)1207 2613 y Fn(and)e Fm(\022)1431 2627 y Fj(2)1505 2613 y Fn(from)f(ev)m(ery)i(p)s(oin)m(t)f(in)h(the)f(other)h (cell.)54 b(Let)35 b Fm(v)3142 2627 y Fj(1)3216 2613 y Fn(b)s(e)f(the)h(v)m(ertex)377 2726 y(in)29 b(one)h(cell)g(of)f(suc)m (h)g(a)h(pair)f(and)f(let)j Fm(v)1725 2740 y Fj(2)1793 2726 y Fn(b)s(e)e(the)g(v)m(ertex)h(in)f(the)h(other)f(cell.)42 b(These)29 b(v)m(ertices)i(are)377 2839 y(joined)e(b)m(y)f(an)h(edge)g (in)g Fm(G)p Fn(,)g(and)f(con)m(tribute)h(\(1)18 b Fl(\000)e Fm(v)2170 2853 y Fj(1)2210 2839 y Fm(v)2254 2853 y Fj(2)2293 2839 y Fn(\))p Fm(=)p Fn(2)30 b(to)g(sdp)2695 2861 y Fk(d)2735 2839 y Fn(.)40 b(Let)29 b Fm(\022)3007 2806 y Fi(0)3055 2839 y Fn(=)c(cos)3273 2806 y Fi(\000)p Fj(1)3382 2839 y Fm(v)3426 2853 y Fj(1)3466 2839 y Fm(v)3510 2853 y Fj(2)3549 2839 y Fn(,)30 b(and)377 2952 y(let)38 b(1)p Fm(=n)f Fn(b)s(e)f(the)h(v)m(olume)g(of)g(eac)m(h)h(cell)g(\(where)f Fm(n)f Fn(is)h(the)g(n)m(um)m(b)s(er)d(of)j(v)m(ertices)i(in)d Fm(G)p Fn(\).)60 b(Then)377 3065 y(ev)m(ery)31 b(pair)f(of)g(p)s(oin)m (ts)g(in)g(the)g(cells)h(mak)m(es)f(an)g(angle)h(of)f Fm(\022)2424 3032 y Fi(0)2466 3065 y Fl(\006)20 b Fm(\015)5 b Fn(,)30 b(and)g(the)g(measure)f(that)h(these)377 3177 y(cells)e(con)m(tribute)g(to)f(sdp)1249 3199 y Fk(c)1311 3177 y Fn(is)f(in)h(the)g(range)g(\(1)13 b Fl(\000)g Fn(cos)r(\()p Fm(\022)2279 3144 y Fi(0)2314 3177 y Fl(\006)g Fm(\015)5 b Fn(\)\))p Fm(=)p Fn(2)p Fm(n)2665 3144 y Fj(2)2706 3177 y Fn(.)39 b(Using)27 b(the)g(fact)h(that)g(cos)16 b Fm(\022)3728 3144 y Fi(0)377 3290 y Fn(di\013ers)27 b(from)e(cos)q(\()p Fm(\022)1059 3257 y Fi(0)1095 3290 y Fl(\006)13 b Fm(\015)5 b Fn(\))27 b(b)m(y)f(at)i(most)e Fl(\006)p Fm(\017)g Fn(\(b)s(ecause)h Fm(\015)32 b Fn(w)m(as)27 b(c)m(hosen)h(to)f(b)s(e)f(small\),)i(w)m(e)f(get)h(that)377 3403 y(the)g(con)m(tribution)g(of)f(con)m(tributing)h(pairs)f(to)h(sdp) 2126 3425 y Fk(c)2187 3403 y Fn(and)f(sdp)2498 3425 y Fk(d)2566 3403 y Fn(are)g(within)g(an)g(additiv)m(e)h(term)f(of)377 3516 y Fm(O)s Fn(\()p Fm(\017)p Fn(\))k(of)g(eac)m(h)g(other.)261 3692 y(4.)46 b Ff(Ne)-5 b(ar)28 b(mixe)-5 b(d)27 b(p)-5 b(airs.)40 b Fn(These)24 b(are)g(pairs)g(of)g(cells)i(that)e(ha)m(v)m (e)i(pairs)d(of)h(p)s(oin)m(ts)g(within)g(distance)h(less)377 3805 y(that)36 b Fm(\022)622 3819 y Fj(1)662 3805 y Fn(,)h(and)d(pairs) i(of)f(p)s(oin)m(ts)h(within)f(distance)h(at)g(least)h Fm(\022)2545 3819 y Fj(1)2584 3805 y Fn(.)56 b(A)m(t)37 b(w)m(orst)f(they)f(con)m(tribute)i(0)377 3918 y(to)43 b(sdp)637 3940 y Fk(d)677 3918 y Fn(.)75 b(T)-8 b(o)43 b(upp)s(er)d(b)s(ound)f(the)k(con)m(tribution)f(to)h(sdp)2443 3940 y Fk(c)2478 3918 y Fn(,)i(consider)d(an)g(arbitrary)f(cell.)77 b(It)377 4031 y(participates)33 b(in)f(mixed)f(pairs)g(only)h(with)g (cells)h(en)m(tirely)g(con)m(tained)g(in)e(the)h(di\013erence)h(of)f(t) m(w)m(o)377 4144 y(caps)f({)g(one)g(of)g(radius)f Fm(\031)23 b Fl(\000)d Fm(\022)1410 4158 y Fj(1)1470 4144 y Fn(+)g Fm(\015)36 b Fn(and)30 b(the)h(other)f(of)h(radius)f Fm(\031)24 b Fl(\000)c Fm(\022)2803 4158 y Fj(1)2862 4144 y Fl(\000)g Fm(\015)5 b Fn(.)42 b(As)30 b(the)h(v)m(olume)g(of)377 4257 y(this)i(region)h(is)f(at)h(most)f(an)g Fm(\017)g Fn(fraction)h(of)f(the)h(v)m(olume)f(of)g(a)g(cap)h(of)f(radius)g Fm(\031)25 b Fl(\000)d Fm(\022)3337 4271 y Fj(1)3409 4257 y Fn(\(and)33 b(the)377 4370 y(v)m(olume)i(of)f(a)h(cap)g(of)g (radius)f Fm(\031)26 b Fl(\000)d Fm(\022)1654 4384 y Fj(2)1727 4370 y Fn(can)35 b(b)s(e)f(neglected\),)k(the)d(in\015uence)f (of)h(the)g(near)f(mixed)377 4483 y(pairs)c(on)h(the)f(v)-5 b(alue)31 b(of)f(sdp)1358 4505 y Fk(c)1423 4483 y Fn(is)h(at)g(most)f (an)g Fm(\017)g Fn(fraction)h(of)g(sdp)2619 4505 y Fk(c)2653 4483 y Fn(.)261 4659 y(5.)46 b Ff(Distant)32 b(mixe)-5 b(d)32 b(p)-5 b(airs.)41 b Fn(These)28 b(are)h(pairs)g(of)f(cells)i (that)f(ha)m(v)m(e)h(pairs)e(of)h(p)s(oin)m(ts)f(within)g(distance)377 4772 y(less)d(than)f Fm(\022)788 4786 y Fj(2)827 4772 y Fn(,)h(and)f(pairs)g(of)g(p)s(oin)m(ts)g(within)g(distance)h(at)g (least)g Fm(\022)2608 4786 y Fj(2)2647 4772 y Fn(.)39 b(A)m(t)25 b(w)m(orst)f(they)h(con)m(tribute)g(0)377 4885 y(to)31 b(sdp)625 4907 y Fk(d)665 4885 y Fn(.)41 b(Their)29 b(con)m(tribution)i(to)g(sdp)1744 4907 y Fk(c)1808 4885 y Fn(can)g(b)s(e)e(b)s(ounded)f(b)m(y)i Fm(\017)p Fn(,)g(as)g(w)m(as)h(done)f(for)f(the)i(case)g(of)377 4998 y(near)g(mixed)e(pairs.)1927 5247 y(9)p eop %%Page: 10 10 10 9 bop 291 -58 a Fn(The)31 b(ab)s(o)m(v)m(e)h(implies)e(sdp)1183 -36 y Fk(d)1250 -58 y Fl(\025)c Fn(sdp)1484 -36 y Fk(c)1539 -58 y Fl(\000)20 b Fm(O)s Fn(\()p Fm(\017)p Fn(\).)43 b(W)-8 b(e)32 b(no)m(w)f(consider)g(opt)2711 -36 y Fk(d)2751 -58 y Fn(.)43 b(Consider)30 b(the)h(maxim)m(um)150 55 y(cut)38 b Fm(C)44 b Fn(in)38 b Fm(G)p Fn(,)h(and)f(let)g(opt)1128 77 y Fk(d)1206 55 y Fn(b)s(e)f(the)h(fraction)h(of)f(edges)g(that)h(it) f(cuts.)63 b(Color)39 b(b)m(y)e(red)h(one)g(side)g(of)150 168 y(the)c(cut)g(and)f(b)m(y)h(green)g(the)g(other)g(side.)51 b(Consider)33 b(no)m(w)g(a)h(cut)g Fm(C)2534 182 y Fk(c)2603 168 y Fn(of)g(the)g(con)m(tin)m(uous)g(sphere,)g(for)150 281 y(whic)m(h)28 b(cells)h(are)f(mono)s(c)m(hromatic)f(and)h(get)h (the)f(color)h(that)f(the)g(cut)g Fm(C)35 b Fn(ga)m(v)m(e)30 b(to)f(the)f(v)m(ertex)h(that)f(lies)150 394 y(within)g(the)g(cell.)41 b(P)m(artitioning)30 b(pairs)e(of)g(cells)h(in)m(to)h(\014v)m(e)e(t)m (yp)s(es)g(as)h(ab)s(o)m(v)m(e,)h(it)e(can)h(b)s(e)e(sho)m(wn)h(that)h (the)150 506 y(fraction)j(of)f(the)g(measure)f(of)i(the)f(edges)h(cut)f (b)m(y)g Fm(C)1958 520 y Fk(c)2024 506 y Fn(is)g(at)h(least)g(opt)2573 528 y Fk(d)2634 506 y Fl(\000)20 b Fm(O)s Fn(\()p Fm(\017)p Fn(\).)44 b(But)31 b(as)g(this)g(fraction)150 619 y(is)f(at)i(most)d (opt)707 641 y Fk(c)741 619 y Fn(,)i(w)m(e)g(get)g(that)g(opt)1410 641 y Fk(d)1476 619 y Fl(\024)25 b Fn(opt)1703 641 y Fk(c)1758 619 y Fn(+)20 b Fm(O)s Fn(\()p Fm(\017)p Fn(\).)291 732 y(W)-8 b(e)32 b(conclude)e(that)1013 894 y(opt)1144 915 y Fk(d)p 1010 934 178 4 v 1010 1018 a Fn(sdp)1147 1039 y Fk(d)1223 955 y Fl(\024)1332 894 y Fn(opt)1463 915 y Fk(c)1518 894 y Fn(+)20 b Fm(O)s Fn(\()p Fm(\017)p Fn(\))p 1329 934 463 4 v 1329 1018 a(sdp)1466 1039 y Fk(c)1521 1018 y Fl(\000)f Fm(O)s Fn(\()p Fm(\017)p Fn(\))1826 955 y Fl(\024)1935 894 y Fn(opt)2066 915 y Fk(c)p 1932 934 172 4 v 1932 1018 a Fn(sdp)2069 1039 y Fk(c)2134 955 y Fn(+)h Fm(O)s Fn(\()p Fm(\017)p Fn(\))25 b Fl(\024)g Fm(\013)c Fn(+)f Fm(O)s Fn(\()p Fm(\017)p Fn(\))p Fm(;)150 1197 y Fn(completing)31 b(the)f(pro)s(of)g(of)g(Theorem)f(4.)117 b Fa(2)150 1439 y Fb(3.2)112 b(A)37 b(Sparse)i(V)-9 b(ersion)150 1611 y Fn(W)h(e)29 b(ha)m(v)m(e)g(seen)g(an)e(in)m(tegralit)m(y)k (ratio)e(arbitrarily)f(close)i(to)f Fm(\013)f Fn(when)f(v)m(ertices)j (are)e(densely)g(pac)m(k)m(ed)h(on)150 1724 y(the)36 b(sphere.)55 b(Namely)-8 b(,)38 b(w)m(e)e(partitioned)g(the)g(sphere)f (in)m(to)h(equal)g(v)m(olume)g(cells)g(of)g(small)f(diameter,)150 1836 y(placed)42 b(a)g(single)h(v)m(ertex)f(in)g(eac)m(h)h(cell,)j(and) 41 b(joined)g(t)m(w)m(o)i(v)m(ertices)g(b)m(y)f(an)g(edge)g(if)g(their) f(distance)150 1949 y(w)m(as)c(close)h(to)f Fm(\022)s Fn(.)58 b(In)35 b(this)i(section)g(w)m(e)g(wish)f(to)h(sho)m(w)f(a)h (similar)f(result)g(when)g(v)m(ertices)i(are)e(placed)150 2062 y(uniformly)28 b(at)i(random)e(on)i(the)g(sphere.)40 b(Moreo)m(v)m(er,)32 b(w)m(e)e(w)m(an)m(t)h(the)f(n)m(um)m(b)s(er)d(of) j(v)m(ertices)h(to)g(b)s(e)e(m)m(uc)m(h)150 2175 y(smaller)f(than)h (the)g(n)m(um)m(b)s(er)d(of)j(cells.)41 b(Sp)s(eci\014cally)-8 b(,)30 b(the)f(graph)f Fm(G)2465 2142 y Fi(0)2517 2175 y Fn(that)h(w)m(e)g(consider)g(in)f(this)h(section)150 2288 y(is)35 b(generated)i(at)f(random)e(as)h(follo)m(ws.)58 b(W)-8 b(e)36 b(let)h(the)e(dimension)f Fm(d)i Fn(b)s(e)f(the)g(same)g (as)h(it)g(w)m(as)g(for)f(the)150 2401 y(dense)j(graph)g Fm(G)f Fn(of)i(the)f(previous)g(section.)65 b(W)-8 b(e)40 b(select)g Fm(n)2265 2368 y Fi(0)2326 2401 y Fn(p)s(oin)m(ts)e (uniformly)e(at)j(random)e(on)h(the)150 2514 y(sphere)c Fm(S)499 2481 y Fk(d)p Fi(\000)p Fj(1)630 2514 y Fn(,)i(where)e(eac)m (h)i(suc)m(h)e(p)s(oin)m(t)h(will)g(represen)m(t)g(a)g(v)m(ertex)h(of)f Fm(G)2731 2481 y Fi(0)2754 2514 y Fn(.)53 b(W)-8 b(e)36 b(connect)g(t)m(w)m(o)g(p)s(oin)m(ts)150 2627 y(b)m(y)d(an)g(edge)g(if) g(the)g(distance)h(\(angle\))g(b)s(et)m(w)m(een)g(them)e(is)g(roughly)h Fm(\022)i Fn(\(namely)-8 b(,)34 b(b)s(et)m(w)m(een)f Fm(\022)3391 2641 y Fj(1)3463 2627 y Fn(and)g Fm(\022)3686 2641 y Fj(2)3725 2627 y Fn(,)150 2740 y(where)d Fm(\022)456 2754 y Fj(1)525 2740 y Fn(and)f Fm(\022)744 2754 y Fj(2)814 2740 y Fn(are)h(c)m(hosen)h(as)f(for)g(the)h(dense)e(graph)h Fm(G)p Fn(\).)41 b(The)29 b(n)m(um)m(b)s(er)f(of)i(v)m(ertices)i Fm(n)3345 2707 y Fi(0)3398 2740 y Fn(is)e(c)m(hosen)150 2853 y(to)41 b(b)s(e)e(suc)m(h)h(that)g(eac)m(h)h(v)m(ertex)g(is)f(exp) s(ected)g(to)h(ha)m(v)m(e)g(degree)g Fm(O)s Fn(\(log)18 b Fm(n)2741 2820 y Fi(0)2764 2853 y Fn(\),)42 b(where)e(the)g Fm(O)j Fn(notation)150 2966 y(hides)28 b(some)g(large)i(constan)m(t.)42 b(W)-8 b(e)30 b(call)g(the)f(resulting)g(graph)f Fm(G)2406 2933 y Fi(0)2458 2966 y Fn(the)h Ff(sp)-5 b(arse)30 b Fn(v)m(ersion.)41 b(W)-8 b(e)30 b(note)f(that)150 3078 y Fm(G)222 3045 y Fi(0)279 3078 y Fn(is)34 b(a)h(random)d(graph,)j(and) f(hence)g(its)h(prop)s(erties)e(\(suc)m(h)h(as)h(the)f(desired)g(in)m (tegralit)m(y)j(ratio\))f(will)150 3191 y(not)30 b(hold)g(with)f (certain)m(t)m(y)-8 b(,)33 b(but)c(only)h(with)g(high)g(probabilit)m(y) g(\(o)m(v)m(er)i(the)e(c)m(hoice)h(of)f Fm(G)3186 3158 y Fi(0)3209 3191 y Fn(\).)41 b(W)-8 b(e)31 b(assume)150 3304 y(throughout)f(that)h Fm(G)894 3271 y Fi(0)947 3304 y Fn(has)f(at)h(least)h(one)f(edge.)291 3417 y(Recall)39 b(that)e(in)f(the)h(dense)g(v)m(ersion)g(w)m(e)g(had)g Fm(n)e Fn(=)h(\()p Fm(O)s Fn(\(1\))p Fm(=\015)5 b Fn(\))2495 3384 y Fk(d)2574 3417 y Fn(v)m(ertices,)41 b(where)36 b Fm(\015)41 b(>)36 b Fn(0)h(is)g(v)m(ery)150 3530 y(small)24 b(and)g(tends)h(to)g(0)g(as)g Fm(d)g Fn(gro)m(ws.)40 b(In)24 b(the)g(sparse)h(v)m(ersion)g(the)g(n)m(um)m(b)s(er)d(of)j(v)m (ertices)i(will)e(b)s(e)f Fm(n)3501 3497 y Fi(0)3549 3530 y Fn(=)h Fm(c)3684 3497 y Fk(d)3725 3530 y Fn(,)150 3643 y(where)h Fm(c)g(>)f Fn(1)i(is)g(some)g(constan)m(t)h(for)f(whic)m (h)f(the)i(exp)s(ected)f(degree)h(b)s(ecomes)e(roughly)g(log)18 b Fm(n)3392 3610 y Fi(0)3440 3643 y Fn(=)25 b(\002\()p Fm(d)p Fn(\).)150 3756 y(F)-8 b(or)30 b(our)g(c)m(hoice)h(of)f Fm(\022)s Fn(,)f(w)m(e)h(will)g(ha)m(v)m(e)h Fm(c)26 b(<)f Fn(2.)41 b(Hence)30 b(the)g(sparse)g(graph)f Fm(G)2759 3723 y Fi(0)2811 3756 y Fn(has)h(m)m(uc)m(h)e(few)m(er)i(v)m(ertices) 150 3869 y(than)d(the)h(dense)f(graph)g Fm(G)p Fn(,)g(giving)h(a)g (sparser)f(con\014guration)h(of)f(p)s(oin)m(ts)g(on)h Fm(S)2887 3836 y Fk(d)p Fi(\000)p Fj(1)3017 3869 y Fn(.)40 b(It)27 b(can)h(b)s(e)f(v)m(eri\014ed)150 3982 y(that)j(for)g(our)f(c)m (hoice)i(of)f(parameters,)g(the)f(a)m(v)m(erage)k(degree)d(in)f(the)h (dense)g(graph)f(is)g Fm(n)3188 3949 y Fj(1)p Fi(\000)p Fk(o)p Fj(\(1\))3436 3982 y Fn(whereas)150 4095 y(the)i(a)m(v)m(erage)j (degree)e(in)f(the)h(sparse)f(graph)f(is)i(\()p Fm(n)1899 4062 y Fi(0)1922 4095 y Fn(\))1957 4062 y Fk(o)p Fj(\(1\))2086 4095 y Fn(,)f(further)f(justifying)h(the)h(terms)e(\\dense")i(and)150 4208 y(\\sparse".)291 4320 y(W)-8 b(e)40 b(wish)e(to)i(sho)m(w)f(that)g (the)g(in)m(tegralit)m(y)j(ratio)e(of)f(the)g(sparse)f(v)m(ersion)i(is) e(similar)h(to)g(that)h(of)150 4433 y(the)34 b(dense)f(v)m(ersion)h (\(in)g(particular,)h(arbitrarily)f(close)h(to)g Fm(\013)p Fn(\).)51 b(Recall)36 b(that)e(sdp)3028 4455 y Fk(d)3102 4433 y Fn(and)f(opt)3413 4455 y Fk(d)3487 4433 y Fn(denote)150 4546 y(the)i(fractional)i(v)-5 b(alues)35 b(of)g(the)g(SDP)g(and)f(of)h (the)g(maxim)m(um)d(cut)j(on)g(the)g(dense)g(graph.)54 b(Similarly)-8 b(,)150 4659 y(w)m(e)33 b(denote)g(b)m(y)f(sdp)846 4681 y Fk(s)915 4659 y Fn(and)f(opt)1224 4681 y Fk(s)1293 4659 y Fn(the)i(fractional)h(v)-5 b(alues)32 b(for)g(the)h(sparse)f (graphs.)45 b(With)33 b(sligh)m(t)g(abuse)150 4772 y(of)40 b(notation,)j(w)m(e)d(consider)g(sdp)1307 4794 y Fk(d)1387 4772 y Fn(and)f(sdp)1710 4794 y Fk(s)1786 4772 y Fn(to)i(b)s(e)e(the)h (fractional)h(v)-5 b(alues)40 b(for)f(the)h(SDP)f(under)g(a)150 4885 y(particular)j(em)m(b)s(edding)d(of)j(the)f(resp)s(ectiv)m(e)i (graph)e(on)g Fm(S)2234 4852 y Fk(d)p Fi(\000)p Fj(1)2365 4885 y Fn(,)j(namely)-8 b(,)44 b(the)e(iden)m(tit)m(y)g(em)m(b)s (edding.)150 4998 y(Hence)f(they)f(do)f(not)h(necessarily)h(corresp)s (ond)e(to)h(optimal)g(solutions)g(to)h(the)f(SDP)-8 b(,)40 b(but)f(they)h(do)1905 5247 y(10)p eop %%Page: 11 11 11 10 bop 150 -58 a Fn(b)s(ound)41 b(the)i(optim)m(um)e(v)-5 b(alue)43 b(of)h(the)f(SDP)f(from)g(b)s(elo)m(w.)79 b(W)-8 b(e)44 b(ha)m(v)m(e)g(seen)f(that)h(also)g(under)d(this)150 55 y(in)m(terpretation,)30 b(opt)883 77 y Fk(d)923 55 y Fm(=)p Fn(sdp)1106 77 y Fk(d)1171 55 y Fl(')25 b Fm(\013)p Fn(.)41 b(W)-8 b(e)28 b(shall)g(sho)m(w)f(that)h(opt)2301 77 y Fk(s)2338 55 y Fm(=)p Fn(sdp)2521 77 y Fk(s)2583 55 y Fl(')d Fm(\013)p Fn(.)40 b(T)-8 b(o)28 b(do)f(this)g(w)m(e)h(sho)m (w)f(that)150 168 y(opt)281 190 y Fk(s)343 168 y Fl(\024)e Fn(opt)571 190 y Fk(d)631 168 y Fn(+)20 b Fm(\017)30 b Fn(and)g(that)h(sdp)1300 190 y Fk(s)1362 168 y Fl(\025)25 b Fn(sdp)1595 190 y Fk(d)1656 168 y Fl(\000)19 b Fm(\017)p Fn(.)150 331 y Fg(Lemma)35 b(7)45 b Ff(F)-7 b(or)34 b(sdp)918 353 y Fk(s)988 331 y Ff(and)f(sdp)1294 353 y Fk(d)1367 331 y Ff(de\014ne)-5 b(d)34 b(as)f(ab)-5 b(ove,)33 b(sdp)2194 353 y Fk(s)2256 331 y Fl(\025)25 b Ff(sdp)2482 353 y Fk(d)2543 331 y Fl(\000)20 b Fm(\017)p Ff(.)291 494 y Fg(Pro)s(of:)40 b Fn(The)28 b(v)-5 b(alues)28 b(of)h(sdp)1305 516 y Fk(d)1373 494 y Fn(and)f(sdp)1684 516 y Fk(s)1749 494 y Fn(are)h(a)m(v)m(erages)h(o)m(v)m(er)g(edges)f(of)f(quan)m (tities)h(\(1)17 b Fl(\000)e Fn(cos)h Fm(\022)3576 461 y Fi(0)3598 494 y Fn(\))p Fm(=)p Fn(2.)150 607 y(Both)26 b(for)g Fm(G)f Fn(and)g Fm(G)847 574 y Fi(0)896 607 y Fn(all)h(edges)h(mak)m(e)f(angles)g(b)s(et)m(w)m(een)h Fm(\022)2147 621 y Fj(1)2211 607 y Fn(and)f Fm(\022)2427 621 y Fj(2)2466 607 y Fn(.)39 b(When)25 b Fm(\022)2833 621 y Fj(1)2898 607 y Fn(and)g Fm(\022)3113 621 y Fj(2)3178 607 y Fn(are)h(su\016cien)m(tly)150 720 y(close,)37 b(the)d(con)m (tribution)h(of)f(eac)m(h)h(edge)g(is)f(\(1)23 b Fl(\000)g Fn(cos)16 b Fm(\022)s Fn(\))p Fm(=)p Fn(2,)35 b(up)e(to)i(an)f(error)g (of)g Fm(\017)p Fn(.)52 b(So)34 b(in)g(b)s(oth)f(cases)150 833 y(the)e(a)m(v)m(erage)i(is)d(roughly)g(\(1)21 b Fl(\000)f Fn(cos)c Fm(\022)s Fn(\))p Fm(=)p Fn(2.)117 b Fa(2)291 946 y Fn(The)38 b(more)f(di\016cult)h(part)g(of)g(the)g(pro)s(of)f(is)h (to)h(sho)m(w)f(that)h(opt)2574 967 y Fk(s)2649 946 y Fl(\024)e Fn(opt)2889 967 y Fk(d)2955 946 y Fn(+)25 b Fm(\017)p Fn(.)63 b(The)38 b(idea)g(is)g(to)150 1059 y(view)e(the)g(sparse)f(graph)h(as)g(a)g(large)h(enough)f(induced)e (subgraph)h(of)h(the)g(dense)f(graph,)i(c)m(hosen)f(at)150 1171 y(random.)57 b(Thereafter,)38 b(w)m(e)f(shall)g(pro)m(v)m(e)g(a)g (general)g(theorem)f(that)h(for)f(ev)m(ery)i(graph,)f(the)g(v)-5 b(alue)37 b(of)150 1284 y(the)30 b(maxim)m(um)c(cut)k(\(as)g(a)g (fraction)g(of)f(the)h(total)h(n)m(um)m(b)s(er)c(of)j(edges\))g(is)g (preserv)m(ed)f(in)g(random)f(large)150 1397 y(enough)40 b(subgraphs)e(of)i(the)g(graph)g(\(up)f(to)i(a)f(negligible)i(error,)g (with)e(high)g(probabilit)m(y)g(o)m(v)m(er)i(the)150 1510 y(c)m(hoice)36 b(of)e(the)g(random)e(subgraph\).)50 b(The)33 b(sparse)h(graph)f Fm(G)2305 1477 y Fi(0)2362 1510 y Fn(can)h(indeed)g(b)s(e)f(view)m(ed)h(as)g(a)h(random)150 1623 y(subgraph)c(of)i(the)g(dense)g(graph)f Fm(G)p Fn(.)48 b(View)33 b(the)g(selection)i(of)e(v)m(ertices)i(to)f(the)f(sparse)f (graph)h(as)g(\014rst)150 1736 y(selecting)27 b(a)f(cell)g(at)g (random,)f(and)g(then)g(selecting)i(a)f(p)s(oin)m(t)f(at)h(random)e(in) h(the)g(cell.)40 b(This)25 b(is)g(iden)m(tical)150 1849 y(to)32 b(the)f(pro)s(cess)g(of)g(selecting)i(a)e(random)e(v)m(ertex)j (from)e(the)h(dense)g(graph.)42 b(\(Recall)33 b(that)f(a)f(v)m(ertex)h (in)150 1962 y(the)j(dense)g(graph)g(is)g(lo)s(cated)i(arbitrarily)e (in)g(a)h(cell,)i(and)c(hence)i(can)f(b)s(e)g(tak)m(en)h(also)h(as)e(a) h(random)150 2075 y(p)s(oin)m(t)28 b(in)g(its)h(cell.\))42 b(F)-8 b(or)29 b(the)f(pro)s(cess)g(of)g(selecting)i(the)f(whole)f (sparse)g(graph,)h(there)f(is)g(also)i(the)e(issue)150 2188 y(of)d(the)h(e\013ect)h(of)e(sampling)g(with)g(replacemen)m(t,)i (but)d(it)i(can)g(b)s(e)e(neglected)j(b)s(ecause)f Fm(n)3152 2155 y Fi(0)3200 2188 y Fl(\034)3316 2122 y(p)p 3392 2122 55 4 v 66 x Fm(n)o Fn(.)39 b(Hence)150 2301 y(the)23 b(sparse)f(graph)g(is)h(just)f(a)g(random)f(sample)h(of)h(the)f(dense)h (graph.)37 b(T)-8 b(o)23 b(simplify)e(the)i(analysis,)i(rather)150 2413 y(than)j(select)i(exactly)g Fm(n)972 2380 y Fi(0)1023 2413 y Fn(v)m(ertices,)h(w)m(e)d(select)i(eac)m(h)g(v)m(ertex)f(of)g (the)f(dense)g(graph)g(indep)s(enden)m(tly)f(with)150 2526 y(probabilit)m(y)k Fm(p)25 b Fn(=)g Fm(n)837 2493 y Fi(0)859 2526 y Fm(=n)p Fn(.)41 b(Hence)31 b(the)f(actual)i(n)m(um)m (b)s(er)c(of)i(v)m(ertices)i(in)d(the)i(sparse)e(graph)h(is)g(a)h (random)150 2639 y(v)-5 b(ariable)31 b(with)f(exp)s(ectation)i Fm(n)1243 2606 y Fi(0)1266 2639 y Fn(,)f(and)e(is)i(strongly)f(concen)m (trated)j(around)c(this)h(exp)s(ectation.)291 2752 y(No)m(w)36 b(w)m(e)f(wish)f(to)h(sho)m(w)g(that)g(b)s(ecause)g(the)g(sparse)f (graph)h(is)f(a)i(random)d(sample)g(of)i(the)g(dense)150 2865 y(graph,)j(the)f(relativ)m(e)h(size)g(of)e(the)h(maxim)m(um)c(cut) k(in)f(the)h(sparse)f(graph)g(is)h(with)f(high)g(probabilit)m(y)150 2978 y(similar)26 b(to)i(that)g(of)f(the)g(dense)f(graph.)39 b(The)27 b(dense)f(graph)h Fm(G)f Fn(has)h(v)m(ery)g(sp)s(ecial)h (structure.)39 b(Ho)m(w)m(ev)m(er,)150 3091 y(in)27 b(the)h(rest)g(of)f (the)h(analysis)g(w)m(e)g(shall)g(ignore)g(most)f(of)h(this)f (structure)g(and)g(treat)i Fm(G)e Fn(as)h(an)f(arbitrary)150 3204 y(graph)32 b(with)g Fm(n)g Fn(v)m(ertices)i(and)d Fm(m)h Fn(edges.)47 b(The)32 b(only)h(prop)s(ert)m(y)e(of)i Fm(G)f Fn(that)h(w)m(e)f(shall)h(use)f(is)g(that)h Fm(G)f Fn(is)150 3317 y Ff(ne)-5 b(arly)36 b(r)-5 b(e)g(gular)35 b Fn({)e(all)h(v)m(ertices)h(in)e Fm(G)f Fn(ha)m(v)m(e)i(roughly)f(the) g(same)g(degree.)49 b(W)-8 b(e)34 b(denote)g(the)f(maxim)m(um)150 3430 y(degree)40 b(b)m(y)g(\001,)h(and)e(assume)g(that)h(\001)g Fl(\024)g Fn(4)p Fm(m=n)p Fn(,)i(i.e.,)h(at)e(most)e(t)m(wice)i(the)e (a)m(v)m(erage)j(degree.)69 b(It)40 b(is)150 3543 y(easy)d(to)f(see)h (that)g(this)f(prop)s(ert)m(y)f(indeed)g(holds)h(for)g(the)g(dense)g (graph.)57 b(As)36 b(for)g(the)g(sparse)f(graph)150 3655 y Fm(G)222 3622 y Fi(0)245 3655 y Fn(,)28 b(w)m(e)g(shall)g(view)g(it)g (as)g(b)s(eing)f(generated)i(from)d Fm(G)h Fn(b)m(y)g(sampling)g(eac)m (h)i(v)m(ertex)g(indep)s(enden)m(tly)d(with)150 3768 y(probabilit)m(y)31 b Fm(p)p Fn(.)41 b(The)30 b(only)h(restriction)h (that)f(w)m(e)g(use)f(here)h(is)g(that)g Fm(p)f Fn(is)h(large)g(enough) g(so)g(that)g(\(with)150 3881 y(high)f(probabilit)m(y\))h(the)g(a)m(v)m (erage)i(degree)e(in)f Fm(G)1796 3848 y Fi(0)1849 3881 y Fn(is)g(signi\014can)m(tly)i(more)e(than)g(log)17 b Fm(n)3073 3848 y Fi(0)3096 3881 y Fn(.)291 3994 y(The)30 b(pro)s(of)g(of)g(the)h(follo)m(wing)h(lemma)d(uses)g(ideas)i(from)e ([9)q(])h(and)g(app)s(ears)g(in)g(the)g(app)s(endix.)150 4157 y Fg(Lemma)35 b(8)45 b Ff(L)-5 b(et)28 b Fm(G)f Ff(b)-5 b(e)27 b(an)g(arbitr)-5 b(ary)30 b(gr)-5 b(aph)29 b(with)f Fm(n)e Ff(vertic)-5 b(es)28 b(and)g Fm(m)f Ff(e)-5 b(dges)27 b(and)i(maximum)f(de)-5 b(gr)g(e)g(e)28 b Fn(\001)150 4270 y Ff(b)-5 b(ounde)g(d)34 b(by)f(twic)-5 b(e)33 b(the)h(aver)-5 b(age)33 b(de)-5 b(gr)g(e)g(e.)44 b(Sele)-5 b(ct)33 b(e)-5 b(ach)34 b(vertex)f(of)g Fm(G)f Ff(indep)-5 b(endently)35 b(with)f(pr)-5 b(ob)g(ability)150 4383 y Fm(p)p Ff(,)41 b(and)f(c)-5 b(onsider)41 b(the)f(r)-5 b(andom)42 b(gr)-5 b(aph)41 b Fm(G)1636 4350 y Fi(0)1699 4383 y Ff(which)f(is)f(the)h(sub) -5 b(gr)g(aph)42 b(of)e Fm(G)f Ff(induc)-5 b(e)g(d)40 b(on)g(the)g(sele)-5 b(cte)g(d)150 4496 y(vertic)g(es.)42 b(Then)33 b(for)h(every)e Fm(\017)25 b(>)h Fn(0)33 b Ff(ther)-5 b(e)33 b(is)g(some)g(c)-5 b(onstant)35 b Fm(c)26 b(>)f Fn(0)33 b Ff(such)g(that)h(if)e Fm(p)25 b Fn(=)h Fm(c)15 b Fn(log)r(\()p Fm(n=)p Fn(\001\))p Fm(=)p Fn(\001)p Ff(,)150 4609 y(then)33 b Fl(j)p Ff(opt)500 4631 y Fk(d)561 4609 y Fl(\000)19 b Ff(opt)774 4631 y Fk(s)811 4609 y Fl(j)26 b Fm(>)f(\017)32 b Ff(with)h(pr)-5 b(ob)g(ability)35 b(at)e(most)g Fm(\017)p Ff(.)42 b(\(The)33 b(pr)-5 b(ob)g(ability)35 b(is)d(taken)h(over)g(the)f(r)-5 b(andom)150 4722 y(choic)g(e)33 b(of)g Fm(G)603 4689 y Fi(0)626 4722 y Ff(.\))291 4885 y Fn(Using)25 b(Lemma)e(7)i(and)f(Lemma)e(8)j(w)m(e)g(conclude)g(that)g (the)g(in)m(tegralit)m(y)j(ratio)d(for)f(the)h(sparse)f(graph)150 4998 y(is)30 b(similar)g(to)h(that)g(of)g(the)f(dense)g(graph,)g (namely)-8 b(,)31 b(arbitrarily)f(close)i(to)f Fm(\013)p Fn(.)1905 5247 y(11)p eop %%Page: 12 12 12 11 bop 150 -58 a Fb(3.3)112 b(Adding)37 b(T)-9 b(riangle)36 b(Constrain)m(ts)150 114 y Fn(In)25 b(this)h(section)i(w)m(e)e(sho)m(w) g(an)g(upp)s(er)f(b)s(ound)f(of)i(roughly)g(0)p Fm(:)p Fn(891)i(on)e(the)g(in)m(tegralit)m(y)j(ratio)f(of)e(the)g(SDP)150 226 y(for)j(MAX)g(CUT)g(with)f(triangle)j(constrain)m(ts.)41 b(The)28 b(triangle)i(constrain)m(ts)g(themselv)m(es)f(are)h(discussed) 150 339 y(in)g(Section)h(4.1.)291 452 y(In)37 b(v)-5 b(alid)37 b(solutions)h(to)g(the)g(GW)g(SDP)-8 b(,)37 b(the)h(unit)f(v)m(ectors)i(are)e(not)h(constrained,)h(in)e(the)h (sense)150 565 y(that)g(they)g(can)g(b)s(e)f(arbitrary)h(p)s(oin)m(ts)f (on)h(the)g(unit)f(sphere.)62 b(But)38 b(when)f(triangle)i(constrain)m (ts)g(are)150 678 y(added,)d(not)f(all)i(p)s(oin)m(t)e (con\014gurations)g(are)h(feasible.)56 b(W)-8 b(e)36 b(sa)m(y)g(that)g(three)f(unit)g(v)m(ectors,)j Fm(u)p Fn(,)f Fm(v)i Fn(and)150 791 y Fm(w)r Fn(,)f(form)d(a)h Ff(forbidden)j(triple)e Fn(if)f(the)h(sum)d(of)i(their)g(inner)f(pro)s (ducts)g Fm(uv)27 b Fn(+)d Fm(v)s(w)j Fn(+)d Fm(w)r(u)35 b(<)g Fl(\000)p Fn(1.)57 b(With)150 904 y(triangle)32 b(constrain)m(ts,)f(only)g(p)s(oin)m(t)f(con\014gurations)h(without)f (forbidden)f(triples)h(are)h(allo)m(w)m(ed.)291 1017 y Fg(Remark:)37 b Fn(With)22 b(the)g(triangle)h(constrain)m(ts,)h(a)f (triple)f(is)f(also)i(forbidden)d(if)i Fl(\000)p Fm(uv)6 b Fl(\000)s Fm(v)s(w)f Fn(+)s Fm(w)r(u)26 b(<)f Fl(\000)p Fn(1.)150 1130 y(This)j(is)h(similar)g(to)g(viewing)h(a)f(p)s(oin)m(t)g (con\014guration)g(as)g(including)g(also)h(ev)m(ery)g(an)m(tip)s(o)s (dal)f(p)s(oin)m(t)g(\(for)150 1243 y Fm(v)36 b Fn(w)m(e)d(include)g (also)h Fl(\000)p Fm(v)s Fn(\))f(and)f(requiring)h(that)g(in)g(the)g (new)f(p)s(oin)m(t)h(con\014guration)g(no)g(triple)g(satis\014es)150 1356 y Fm(uv)25 b Fn(+)d Fm(v)s(w)j Fn(+)c Fm(w)r(u)30 b(<)f Fl(\000)p Fn(1.)49 b(Note)34 b(that)g(a)f(forbidden)e(triple)i (cannot)h(con)m(tain)g(t)m(w)m(o)g(p)s(oin)m(ts)f(an)m(tip)s(o)s(dal)g (to)150 1468 y(eac)m(h)40 b(other.)67 b(T)-8 b(ec)m(hnically)g(,)44 b(w)m(e)c(should)e(pro)m(v)m(e)i(Prop)s(osition)f(10)h(in)e(the)i (framew)m(ork)e(that)h(includes)150 1581 y(an)m(tip)s(o)s(dal)h(p)s (oin)m(ts.)67 b(It)40 b(is)f(easy)h(to)g(see)g(that)g(the)f(pro)s(of)g (extends)g(to)h(this)g(case)g(with)f(virtually)h(no)150 1694 y(c)m(hange.)291 1807 y(The)c(in)m(tegralit)m(y)i(ratio)f (examples)f(sho)m(wn)f(for)h(the)g(GW)h(SDP)f(con)m(tain)h(forbidden)e (triples,)j(and)150 1920 y(hence)21 b(are)g(infeasible)h(when)d (triangle)k(constrain)m(ts)e(are)h(presen)m(t.)37 b(The)21 b(sparse)f(v)m(ersion)h(w)m(as)h(in)m(tro)s(duced)150 2033 y(so)33 b(as)g(to)h(o)m(v)m(ercome)g(this)f(problem.)46 b(If)33 b(a)g(random)e(p)s(oin)m(t)i(con\014guration)g(is)g(sparse)f (enough,)i(then)e(it)150 2146 y(con)m(tains)39 b(no)f(forbidden)f (triple.)64 b(Unfortunately)-8 b(,)42 b(in)37 b(this)h(case)i(it)e (will)h(also)g(not)f(con)m(tain)i(pairs)e(of)150 2259 y(p)s(oin)m(ts)27 b(at)h(distance)g(more)e(than)h Fm(\022)j 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y(con\014guration)f(that)g(con)m(tains)h(a)f(small)f(n)m(um)m(b)s (er)e(of)j(forbidden)e(triples.)41 b(This)30 b(will)h(allo)m(w)h(us)e (to)h(pac)m(k)150 3049 y(more)g(p)s(oin)m(ts)h(on)g(the)g(sphere,)g (giving)h(a)g(higher)e(v)-5 b(alue)33 b(for)f Fm(\014)5 b Fn(.)46 b(W)-8 b(e)33 b(will)f(still)h(need)f(to)h(get)g(rid)f(of)g (the)150 3162 y(forbidden)d(triples,)i(but)e(this)i(will)f(turn)g(out)g (to)h(b)s(e)f(easy)-8 b(,)32 b(b)s(ecause)e(their)g(n)m(um)m(b)s(er)f (is)h(small.)291 3275 y(Before)43 b(pro)s(ceeding)f(with)g(a)g (detailed)h(pro)s(of,)i(let)e(us)e(in)m(tro)s(duce)h(a)h(kno)m(wn)e (tec)m(hnical)j(lemma.)150 3388 y(Its)37 b(pro)s(of)g(is)g(presen)m (ted)h(for)f(completeness)g(\(and)h(also)g(b)s(ecause)f(w)m(e)h(could)g (not)f(\014nd)f(a)i(con)m(v)m(enien)m(t)150 3501 y(reference)31 b(for)f(this)g(lemma\).)150 3688 y Fg(Lemma)35 b(9)45 b Ff(F)-7 b(or)25 b(al)5 b(l)24 b Fm(\031)s(=)p Fn(2)i Fm(<)f(\014)31 b(<)25 b(\031)h Ff(and)f(two)g(r)-5 b(andom)26 b(unit)d(ve)-5 b(ctors)25 b(in)e Fm(R)2724 3655 y Fk(d)2764 3688 y Ff(,)j(the)e(pr)-5 b(ob)g(ability)25 b(that)g(their)150 3801 y(distanc)-5 b(e)36 b(\(namely,)h(the)e(angle)g(b)-5 b(etwe)g(en)36 b(them\))f(is)g(at)h(le)-5 b(ast)36 b Fm(\014)k Ff(is)35 b(r)-5 b(oughly)36 b Fn(\(sin)15 b Fm(\014)5 b Fn(\))3060 3768 y Fk(d)p Fi(\000)p Fj(1)3191 3801 y Ff(.)49 b(Equivalently)150 3914 y(\(using)42 b Fn(sin)547 3879 y Fj(2)602 3914 y Fm(\014)32 b Fn(+)27 b(cos)904 3881 y Fj(2)959 3914 y Fm(\014)47 b Fn(=)42 b(1)p Ff(\),)j(for)d(al)5 b(l)43 b Fn(0)f Fm(<)g(a)g(<)g Fn(1)g Ff(and)h(two)g(r)-5 b(andom)44 b(unit)e(ve)-5 b(ctors)43 b(in)e Fm(R)3519 3881 y Fk(d)3560 3914 y Ff(,)j(the)150 4027 y(pr)-5 b(ob)g(ability)27 b(that)g(their)f(inner)f(pr)-5 b(o)g(duct)27 b(is)f(at)f(le)-5 b(ast)27 b Fm(a)e Ff(is)g(r)-5 b(oughly)27 b Fn(\(1)t Fl(\000)t Fm(a)2590 3994 y Fj(2)2630 4027 y Fn(\))2665 3994 y Fj(\()p Fk(d)p Fi(\000)p Fj(1\))p Fk(=)p Fj(2)2921 4027 y Ff(.)39 b(Fixing)26 b(the)f(dir)-5 b(e)g(ction)150 4140 y(of)39 b(one)f(of)h(the)f(two)i(ve)-5 b(ctors)39 b(to)g(b)-5 b(e)38 b(along)h Fm(x)1727 4154 y Fj(1)1767 4140 y Ff(,)g(we)g(state)g(the)f(lemma)i(in)e(yet)g (another)j(form)e(\(which)150 4253 y(also)34 b(quanti\014es)f(the)g (term)g(\\r)-5 b(oughly"\),)34 b(namely,)673 4412 y Fm(c)p 631 4452 124 4 v 631 4469 a Fl(p)p 706 4469 48 4 v 706 4547 a Fm(d)764 4473 y Fn(\(1)21 b Fl(\000)f Fm(a)1004 4436 y Fj(2)1043 4473 y Fn(\))1088 4407 y Fh(d)p Fd(\000)p Fc(1)p 1088 4421 111 4 v 1128 4462 a(2)1238 4473 y Fl(\024)25 b Fm(\026)p Fn(\()p Fl(f)p Fn(\()p Fm(x)1556 4487 y Fj(1)1597 4473 y Fm(;)15 b(:)g(:)g(:)h(;)f(x)1850 4488 y Fk(d)1891 4473 y Fn(\))26 b Fl(2)e Fm(S)2098 4436 y Fk(d)p Fi(\000)p Fj(1)2229 4473 y Fn(;)48 b Fm(x)2354 4487 y Fj(1)2419 4473 y Fm(>)25 b(a)p Fl(g)p Fn(\))h Fl(\024)2775 4412 y Fn(1)p 2775 4452 46 4 v 2775 4536 a(2)2830 4473 y(\(1)21 b Fl(\000)f Fm(a)3070 4436 y Fj(2)3109 4473 y Fn(\))3154 4407 y Fh(d)p Fd(\000)p Fc(1)p 3155 4421 111 4 v 3195 4462 a(2)3634 4473 y Fn(\(2\))150 4698 y Ff(for)33 b(some)h(c)-5 b(onstant)34 b Fm(c)p Ff(.)291 4885 y Fg(Pro)s(of:)51 b Fn(Let)35 b Fm(\014)40 b Fn(b)s(e)35 b(the)g(angle)h(suc)m(h)e(that)i (cos)16 b Fm(\014)38 b Fn(=)33 b Fm(a)p Fn(.)54 b(W)-8 b(e)36 b(are)g(in)m(terested)g(in)e(the)i(ratio)g(of)f(the)150 4998 y(area)26 b(of)g(the)g(cap)g(of)f(radius)g Fm(\014)31 b Fn(to)26 b(the)g(area)g(of)g(the)f(whole)h(sphere.)39 b(Cut)24 b(o\013)i(this)g(cap)g(from)e(the)h(sphere.)1905 5247 y(12)p eop %%Page: 13 13 13 12 bop 150 -58 a Fn(This)27 b(cap)g(no)m(w)h(has)f(a)g(basis)h(of)f (radius)g(sin)14 b Fm(\014)33 b Fn(and)27 b(w)m(e)g(are)h(in)m (terested)h(in)e(the)g(surface)g(area)i(of)e(the)h(cap)150 55 y(without)34 b(the)h(basis.)52 b(This)33 b(can)h(b)s(e)g(upp)s(er)e (b)s(ounded)g(b)m(y)i(the)g(surface)h(area)g(of)f(a)g(half)h(sphere)e (in)h Fm(R)3710 22 y Fk(d)150 168 y Fn(of)g(radius)f(sin)15 b Fm(\014)5 b Fn(.)52 b(\(F)-8 b(ormally)g(,)36 b(this)e(can)g(b)s(e)f 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2658 y Fj(2)234 2644 y Fn(,)d(and)f(compute)f(the)i(probabilit)m(y)g Fm(p)1513 2658 y Fj(1)1552 2644 y Fn(\()p Fm(a)p Fn(\))g(that)h Fl(\000)p Fm(a)28 b Fl(\025)h Fm(v)2194 2658 y Fj(1)2233 2644 y Fm(v)2277 2658 y Fj(2)2345 2644 y Fm(>)g Fl(\000)p Fm(a)21 b Fl(\000)h Fn(1)p Fm(=k)s Fn(.)48 b(Rather)32 b(than)h(compute)150 2757 y Fm(p)196 2771 y Fj(1)235 2757 y Fn(\()p Fm(a)p Fn(\))k(exactly)-8 b(,)40 b(w)m(e)d(upp)s(er)e(b) s(ound)f(it)j(b)m(y)f(the)h(probabilit)m(y)f(that)h Fm(v)2530 2771 y Fj(1)2570 2757 y Fm(v)2614 2771 y Fj(2)2689 2757 y Fl(\024)d(\000)p Fm(a)p Fn(,)k(or)f(equiv)-5 b(alen)m(tly)d(,)40 b(the)150 2869 y(probabilit)m(y)31 b(that)g Fm(v)856 2883 y Fj(1)895 2869 y Fm(v)939 2883 y Fj(2)1004 2869 y Fm(>)25 b(a)p Fn(.)41 b(F)-8 b(rom)30 b(Lemma)e(9)j(w)m(e)g(get)g (that)1517 3066 y Fm(p)1563 3080 y Fj(1)1603 3066 y Fn(\()p Fm(a)p Fn(\))26 b Fl(\024)1853 3005 y Fn(1)p 1853 3045 46 4 v 1853 3129 a(2)1908 3066 y(\(1)21 b Fl(\000)f Fm(a)2148 3029 y Fj(2)2187 3066 y Fn(\))2232 3000 y Fh(d)p Fd(\000)p Fc(1)p 2233 3014 111 4 v 2273 3055 a(2)2357 3066 y Fm(:)150 3251 y Fn(No)m(w)26 b(conditioned)g(on)g(the)f(inner)g(pro)s(duct)f Fm(v)1718 3265 y Fj(1)1758 3251 y Fm(v)1802 3265 y Fj(2)1867 3251 y Fn(b)s(eing)h(in)g(the)h(range)g(as)g(ab)s(o)m(v)m(e,)i(w)m(e)e (c)m(ho)s(ose)g(the)g(third)150 3364 y(v)m(ector)d Fm(v)458 3378 y Fj(3)519 3364 y Fn(at)f(random)e(and)g(compute)h(the)h (probabilit)m(y)g Fm(p)2123 3378 y Fj(2)2162 3364 y Fn(\()p Fm(a)p Fn(\))g(that)g Fm(v)2534 3378 y Fj(3)2573 3364 y Fn(\()p Fm(v)2652 3378 y Fj(1)2695 3364 y Fn(+)r Fm(v)2812 3378 y Fj(2)2851 3364 y Fn(\))k Fm(<)f Fl(\000)p Fn(1)r(+)r Fm(a)r Fn(+)r(1)p Fm(=k)h Fn(\(whic)m(h)150 3477 y(corresp)s(onds)c(to) i(the)f(ev)m(en)m(t)i(that)f(the)g(triple)f(is)h(forbidden\).)37 b(T)-8 b(aking)24 b Fm(k)i Fn(large)f(enough,)f(w)m(e)g(will)g(b)s(e)e (able)150 3590 y(to)31 b(neglect)g(the)f(term)f(1)p Fm(=k)34 b Fn(compared)29 b(to)i(the)f(v)-5 b(alue)30 b(of)g Fm(a)p 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Fm(v)845 4055 y Fj(3)910 4041 y Fn(has)g(inner)f(pro)s(duct)g(more)g(negativ)m(e)k(than)d(\()p Fl(\000)p Fn(1)10 b(+)g Fm(a)p Fn(\))p Fm(=)2783 3970 y Fl(p)p 2859 3970 V 71 x Fn(2)20 b Fl(\000)g Fn(2)p Fm(a)26 b Fn(=)f Fl(\000)3301 3968 y Fe(p)p 3384 3968 367 4 v 73 x Fn(\(1)c Fl(\000)f Fm(a)p Fn(\))p Fm(=)p Fn(2)150 4154 y(with)30 b(a)h(unit)f(v)m(ector.)42 b(Using)31 b(Lemma)d(9)j(w)m(e)g(upp)s(er)d(b)s(ound)g(this)j(probabilit)m(y)g(b)m (y)1141 4351 y Fm(p)1187 4365 y Fj(2)1226 4351 y Fn(\()p Fm(a)p Fn(\))26 b Fl(\024)1476 4290 y Fn(1)p 1476 4330 46 4 v 1476 4414 a(2)1531 4351 y(\(1)21 b Fl(\000)1733 4290 y Fn(1)g Fl(\000)f Fm(a)p 1733 4330 205 4 v 1813 4414 a Fn(2)1948 4351 y(\))1993 4285 y Fh(d)p Fd(\000)p Fc(1)p 1993 4299 111 4 v 2033 4340 a(2)2143 4351 y Fn(=)2249 4290 y(1)p 2249 4330 46 4 v 2249 4414 a(2)2304 4351 y(\()2349 4290 y(1)h(+)f Fm(a)p 2349 4330 205 4 v 2429 4414 a Fn(2)2564 4351 y(\))2609 4285 y Fh(d)p Fd(\000)p Fc(1)p 2609 4299 111 4 v 2649 4340 a(2)2734 4351 y Fm(:)150 4541 y Fn(No)m(w)33 b(w)m(e)g(need)f(to)h(sum)d Fm(p)1062 4555 y Fj(1)1102 4541 y Fn(\()p Fm(a)p Fn(\))p Fm(p)1266 4555 y Fj(2)1305 4541 y Fn(\()p Fm(a)p Fn(\))j(o)m(v)m(er)h(all)f Fm(a)p Fn(.)47 b(Let)33 b(us)e(\014rst)h(\014nd)f(the)h(v)-5 b(alue)33 b(of)g Fm(a)f Fn(that)h(maximizes)150 4654 y(the)h(upp)s(er)e(b)s(ound)f(on)j(the)g(pro)s(duct)f Fm(p)1538 4668 y Fj(1)1577 4654 y Fn(\()p Fm(a)p Fn(\))p Fm(p)1741 4668 y Fj(2)1781 4654 y Fn(\()p Fm(a)p Fn(\).)51 b(The)34 b(expression)f(\(1)24 b Fl(\000)e Fm(a)2852 4621 y Fj(2)2891 4654 y Fn(\)\(1)i(+)e Fm(a)p Fn(\))34 b(has)g(a)g(unique)150 4767 y(maxim)m(um)27 b(at)k Fm(a)26 b Fn(=)f(1)p Fm(=)p Fn(3.)42 b(This)29 b(giv)m(es:)1201 4969 y Fm(p)1247 4983 y Fj(1)1286 4969 y Fn(\()p Fm(a)p Fn(\))p Fm(p)1450 4983 y Fj(2)1490 4969 y Fn(\()p Fm(a)p Fn(\))d Fl(\024)1740 4907 y Fn(1)p 1740 4948 46 4 v 1740 5031 a(4)1795 4969 y(\()1840 4907 y(8)p 1840 4948 V 1840 5031 a(9)1916 4969 y Fl(\001)1971 4907 y Fn(2)p 1971 4948 V 1971 5031 a(3)2027 4969 y(\))2072 4903 y Fh(d)p Fd(\000)p Fc(1)p 2072 4917 111 4 v 2112 4957 a(2)2222 4969 y Fn(=)2328 4907 y(1)p 2328 4948 46 4 v 2328 5031 a(4)2383 4969 y(\()2428 4907 y(16)p 2428 4948 91 4 v 2428 5031 a(27)2530 4969 y(\))2575 4903 y Fh(d)p Fd(\000)p Fc(1)p 2575 4917 111 4 v 2615 4957 a(2)1905 5247 y Fn(13)p eop %%Page: 14 14 14 13 bop 150 -58 a Fn(where)27 b(the)g(base)g(of)h(the)f(exp)s(onen)m (t)g(is)g(acurate)h(up)e(to)i(an)f(additiv)m(e)i(error)d(of)i Fm(O)s Fn(\(1)p Fm(=k)s Fn(\).)41 b(Summing)24 b(o)m(v)m(er)150 55 y Fm(k)36 b Fn(v)-5 b(alues)33 b(of)g Fm(a)p Fn(,)h(and)e(absorbing) h(the)g(leading)g(term)f Fm(k)s(=)p Fn(4)i(in)f(the)g(base)g(of)g(the)g (exp)s(onen)m(t)g(\(whic)m(h)g(can)150 168 y(b)s(e)d(done)g(with)g (little)i(e\013ect)g(when)d Fm(d)i Fn(is)f(su\016cien)m(tly)i(large\),) g(the)e(prop)s(osition)g(is)h(pro)m(v)m(ed.)116 b Fa(2)291 281 y Fn(Our)25 b(plan)h(is)h(to)g(place)g Fm(n)f Fn(p)s(oin)m(ts)h(at) g(random)d(on)j Fm(S)2066 248 y Fk(d)p Fi(\000)p Fj(1)2223 281 y Fn(and)e(ha)m(v)m(e)j(that)f(the)g(n)m(um)m(b)s(er)d(of)i (forbidden)150 394 y(triples)31 b(b)s(e)e Fm(O)s Fn(\()p Fm(\017n)p Fn(\).)41 b(Later)31 b(w)m(e)g(will)f(deal)h(with)f(the)h (forbidden)e(triples.)150 552 y Fg(Corollary)k(11)46 b Ff(F)-7 b(or)43 b(an)g(arbitr)-5 b(arily)45 b(smal)5 b(l)43 b(c)-5 b(onstant)44 b Fn(0)g Fm(<)e(\017)h(<)f Fn(1)p Ff(,)j(for)e(lar)-5 b(ge)43 b(enough)g Fm(d)p Ff(,)i(if)d Fm(n)g Fl(')150 675 y Fn(\()195 584 y Fi(p)p 254 584 71 3 v 55 x Fj(27)p 195 654 130 4 v 242 707 a(4)354 675 y Fl(\000)20 b Fm(\017)p Fn(\))517 642 y Fk(d=)p Fj(2)660 675 y Ff(p)-5 b(oints)34 b(ar)-5 b(e)33 b(plac)-5 b(e)g(d)35 b(at)e(r)-5 b(andom)35 b(on)d Fm(S)1981 642 y Fk(d)p Fi(\000)p Fj(1)2112 675 y Ff(,)g(the)h(exp)-5 b(e)g(cte)g(d)34 b(numb)-5 b(er)33 b(of)f(forbidden)i(triples)150 788 y(is)f(at)g(most)g Fm(\017n)p Ff(.)291 947 y Fg(Pro)s(of:)61 b Fn(With)41 b Fm(n)e Fn(random)g(p)s(oin)m(ts)h(in)g Fm(S)1777 914 y Fk(d)p Fi(\000)p Fj(1)1907 947 y Fn(,)j(the)d(exp)s (ected)h(n)m(um)m(b)s(er)d(of)i(forbidden)f(triples)h(is)150 991 y Fe(\000)188 1020 y Fk(n)192 1091 y Fj(3)231 991 y Fe(\001)269 1060 y Fn(\()314 1024 y Fj(16)p 314 1039 71 4 v 314 1091 a(27)405 1060 y Fn(+)10 b Fm(\017)523 1027 y Fi(0)546 1060 y Fn(\))581 1027 y Fk(d=)p Fj(2)692 1060 y Fn(,)27 b(where)d Fm(\017)1038 1027 y Fi(0)1087 1060 y Fn(is)h(carried)h(o)m(v)m(er)g(from)e(Prop)s(osition)h(10)i (\(where)e(it)h(w)m(as)f(denoted)h(b)m(y)f Fm(\017)p Fn(\),)i(and)150 1182 y(can)e(b)s(e)f(made)f(m)m(uc)m(h)h(smaller)g (than)g Fm(\017)g Fn(b)m(y)h(making)f Fm(d)g Fn(su\016cien)m(tly)i (large.)39 b(T)-8 b(aking)26 b Fm(n)e Fl(')3162 1117 y(p)p 3238 1117 37 4 v 65 x Fm(\017)p Fn(\()3320 1147 y Fj(16)p 3320 1162 71 4 v 3320 1214 a(27)3409 1182 y Fn(+)8 b Fm(\017)3525 1149 y Fi(0)3549 1182 y Fn(\))3584 1149 y Fi(\000)p Fk(d=)p Fj(4)150 1295 y Fn(giv)m(es)32 b(the)e(desired)g(result.)117 b Fa(2)291 1408 y Fn(No)m(w)34 b(w)m(e)f(construct)h(a)f(graph)g Fm(G)f Fn(b)m(y)h(placing)h Fm(n)e Fn(p)s(oin)m(ts)h(at)h(random)d(on)i Fm(S)2931 1375 y Fk(d)p Fi(\000)p Fj(1)3062 1408 y Fn(,)h(where)e(the)h(v)-5 b(alue)150 1521 y(of)42 b Fm(n)e Fn(is)i(as)g(indicated)g(in)f (Corollary)h(11,)j(and)c(joining)g(b)m(y)h(an)f(edge)h(pairs)f(of)h(p)s (oin)m(ts)f(that)h(are)g(of)150 1634 y(distance)29 b(at)f(least)i Fm(\014)5 b Fn(.)40 b(\(Remark:)f(as)28 b(argued)g(for)g(the)g(dense)g (graph,)g(when)f(the)h(dimension)e Fm(d)j Fn(is)f(large)150 1747 y(enough)36 b(then)h(with)f(o)m(v)m(erwhelming)h(probabilit)m(y)g (all)g(but)f(a)h(negligible)i(fraction)e(of)g(the)f(edges)i(will)150 1860 y(happ)s(en)33 b(to)i(connect)h(pairs)e(of)h(p)s(oin)m(ts)g(whose) f(distance)h(is)g(essen)m(tially)i(equal)e(to)g Fm(\014)5 b Fn(.\))54 b(W)-8 b(e)36 b(w)m(an)m(t)g(the)150 1973 y(a)m(v)m(erage)29 b(degree)f(in)f Fm(G)f Fn(to)h(b)s(e)g(at)g(least)h (log)18 b Fm(n)p Fn(.)39 b(Using)27 b(the)g(b)s(ounds)e(of)i(Lemma)e(9) i(and)f(taking)i(the)f(limit)150 2086 y(when)h Fm(\017)d Fl(\000)-15 b(!)25 b Fn(0,)30 b(w)m(e)g(c)m(ho)s(ose)g(\(sin)15 b Fm(\014)5 b Fn(\))1393 2053 y Fj(2)1462 2086 y Fn(to)30 b(b)s(e)f(arbitrarily)g(close)h(to)g(4)p Fm(=)2547 2011 y Fl(p)p 2624 2011 91 4 v 2624 2086 a Fn(27)q(,)f(implying)f Fm(\014)j Fl(')25 b Fn(118)p Fm(:)p Fn(67)32 b(and)150 2199 y(cos)16 b Fm(\014)33 b Fl(')c(\000)p Fn(0)p Fm(:)p Fn(4798.)48 b(The)32 b(graph)g Fm(G)f Fn(can)i(b)s(e)e(view)m(ed)i(as)g (a)f(sparse)g(v)m(ersion)h(of)f(a)h(dense)f(graph)g(similar)150 2312 y(to)d(the)g(construction)g(of)g(Section)h(3.2,)g(with)e(the)h (only)g(di\013erence)g(that)g Fm(\014)34 b Fn(replaces)c Fm(\022)s Fn(.)39 b(Hence)29 b(for)g(the)150 2424 y(SDP)h(of)h(GW)g(on) f Fm(G)g Fn(w)m(e)h(ha)m(v)m(e)g(an)f(in)m(tegralit)m(y)k(ratio)d(of)g (roughly)1735 2559 y(2)p Fm(\014)p 1548 2599 476 4 v 1548 2683 a(\031)s Fn(\(1)21 b Fl(\000)f Fn(cos)c Fm(\014)5 b Fn(\))2059 2620 y Fl(')25 b Fn(0)p Fm(:)p Fn(891)291 2822 y(T)-8 b(o)36 b(extend)f(this)g(result)h(to)g(the)f(SDP)g(with)g 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Fn(w)m(as)g(nearly)h(regular,)h (this)e(pro)s(cess)g(e\013ects)h(only)g(an)f Fm(O)s Fn(\()p Fm(\017)p Fn(\))g(fraction)h(of)g(the)150 3613 y(edges)32 b(from)f Fm(G)p Fn(.)44 b(Hence)33 b(the)f(in)m(tegralit)m(y)i(ratio)f (c)m(hanges)g(only)f(b)m(y)f Fm(O)s Fn(\()p Fm(\017)p Fn(\).)46 b(This)31 b(completes)h(the)g(pro)s(of)150 3725 y(of)f(Theorem)e(3.)291 3838 y Fg(Remark:)39 b Fn(Rather)25 b(than)g(mo)s(dify)f(the)i(solution)f(to)i(the)e(SDP)-8 b(,)26 b(w)m(e)g(could)f(mo)s(dify)f Fm(G)h Fn(b)m(y)g(remo)m(ving)150 3951 y(from)35 b(eac)m(h)j(forbidden)d(triple)i(one)g(v)m(ertex.)61 b(The)36 b(results)h(regarding)g(the)g(in)m(tegralit)m(y)i(ratio)f (remain)150 4064 y(unc)m(hanged,)30 b(but)g(the)h(description)f(of)g (the)h(graph)f Fm(G)g Fn(b)s(ecomes)f(a)i(bit)f(more)g(complicated.)150 4343 y Fo(4)137 b(The)46 b(Appro)l(ximation)g(Ratio)150 4546 y Fn(In)26 b(this)h(section)h(w)m(e)g(sho)m(w)f(that)h(ev)m(en)f (the)h(b)s(est)e(h)m(yp)s(erplane)g(do)s(es)h(not)g(giv)m(e)i(an)e 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(ed)e(b)s(elo)m(w)i(seem)e(promis-)150 339 y(ing.)38 b(One)22 b(w)m(a)m(y)i(of)f(viewing)g(them)e(is)i(as)g(the)g(set)g(of)g (v)-5 b(alid)23 b(constrain)m(ts)g(that)h(in)m(v)m(olv)m(e)g(triples)f (of)g(v)m(ertices.)150 452 y(Regardless)31 b(of)g(whic)m(h)g Fl(\006)p Fn(1)g(v)-5 b(alues)31 b(the)g(three)g(v)m(ertices)h(receiv)m (e,)h(the)e(follo)m(wing)i(constrain)m(t)f(is)e(alw)m(a)m(ys)150 565 y(satis\014ed:)1578 678 y Fl(j)p Fm(v)1647 692 y Fk(i)1696 678 y Fn(+)20 b Fm(v)1831 692 y Fk(j)1888 678 y Fn(+)f Fm(v)2022 693 y Fk(k)2065 678 y Fl(j)2090 641 y Fj(2)2155 678 y Fl(\025)25 b Fn(1)p Fm(:)150 845 y Fn(F)-8 b(or)31 b(unit)f(v)m(ectors,)i(this)e(giv)m(es:)1481 958 y Fm(v)1525 972 y Fk(i)1553 958 y Fm(v)1597 972 y Fk(j)1654 958 y Fn(+)20 b Fm(v)1789 972 y Fk(j)1825 958 y Fm(v)1869 973 y Fk(k)1932 958 y Fn(+)g Fm(v)2067 973 y Fk(k)2110 958 y Fm(v)2154 972 y Fk(i)2207 958 y Fl(\025)25 b(\000)p Fn(1)1215 b(\(3\))150 1125 y(W)-8 b(e)30 b(ma)m(y)f(add)f(a)h (similar)g(constrain)m(t)h(when)e Fl(\000)p Fm(v)1825 1139 y Fk(j)1890 1125 y Fn(is)h(substituted)f(for)h Fm(v)2636 1139 y Fk(j)2673 1125 y Fn(,)g(as)g(the)g(v)-5 b(alue)30 b(of)f Fl(\000)p Fm(v)3443 1139 y Fk(j)3508 1125 y Fn(is)g(also)150 1238 y(still)i Fl(\006)p Fn(1.)41 b(This)30 b(giv)m(es)h(the)g(follo)m (wing)h(constrain)m(t.)1445 1442 y Fl(\000)20 b Fm(v)1580 1456 y Fk(i)1608 1442 y Fm(v)1652 1456 y Fk(j)1709 1442 y Fl(\000)g Fm(v)1844 1456 y Fk(j)1881 1442 y Fm(v)1925 1457 y Fk(k)1988 1442 y Fn(+)g Fm(v)2123 1457 y Fk(k)2165 1442 y Fm(v)2209 1456 y Fk(i)2263 1442 y Fl(\025)25 b(\000)p Fn(1)1159 b(\(4\))150 1646 y(Constrain)m(t)31 b(4)g(can)f(b)s(e)g(seen) g(to)i(b)s(e)d(equiv)-5 b(alen)m(t)32 b(to)1293 1850 y Fl(j)p Fm(v)1362 1864 y Fk(i)1410 1850 y Fl(\000)20 b Fm(v)1545 1864 y Fk(j)1582 1850 y Fl(j)1607 1813 y Fj(2)1667 1850 y Fn(+)g Fl(j)p Fm(v)1827 1864 y Fk(j)1884 1850 y Fl(\000)g Fm(v)2019 1865 y Fk(k)2061 1850 y Fl(j)2086 1813 y Fj(2)2151 1850 y Fl(\025)25 b(j)p Fm(v)2316 1864 y Fk(i)2365 1850 y Fl(\000)20 b Fm(v)2500 1865 y Fk(k)2543 1850 y Fl(j)2568 1813 y Fj(2)3634 1850 y Fn(\(5\))150 2055 y(using)30 b(the)g(fact)i(that)e Fl(j)p Fm(u)21 b Fl(\000)f Fm(v)s Fl(j)1181 2022 y Fj(2)1246 2055 y Fn(=)25 b Fm(u)1394 2022 y Fj(2)1453 2055 y Fn(+)20 b Fm(v)1591 2022 y Fj(2)1651 2055 y Fl(\000)g Fn(2)p Fm(uv)29 b Fn(=)c(2)20 b Fl(\000)g Fn(2)p Fm(uv)34 b Fn(for)c(unit)g(v)m (ectors.)42 b(Constrain)m(t)31 b(5)f(has)h(a)150 2167 y(con)m(v)m(enien)m(t)e(geometric)f(in)m(terpretation,)i(sa)m(ying)e (that)f(the)g(angle)h(op)s(ened)f(b)m(y)g(three)g(v)m(ertices)i(is)e (nev)m(er)150 2280 y(more)k(than)h Fm(\031)s(=)p Fn(2)g(\(the)h(case)g (of)f Fm(\031)s(=)p Fn(2)g(giv)m(es)i(equalit)m(y)f(in)f(the)g (constrain)m(t,)h(b)m(y)f(Pythagoras')h(theorem\).)150 2393 y(W)-8 b(e)33 b(call)g(this)f(the)f Ff(acute)j(angle)g(rule)p Fn(.)45 b(W)-8 b(e)33 b(note)g(that)f(from)e(constrain)m(t)j(3)f(w)m(e) g(can)g(deriv)m(e)h(a)f(triangle)150 2506 y(inequalit)m(y)g(similar)d (to)j(constrain)m(t)f(5,)g(but)f(with)g Fm(v)1928 2520 y Fk(j)1995 2506 y Fn(replaced)h(b)m(y)f Fl(\000)p Fm(v)2595 2520 y Fk(j)2631 2506 y Fn(.)291 2619 y(Recall)38 b(that)f(the)g(in)m (tegralit)m(y)i(gap)e(for)f(the)h(5-cycle)i(w)m(as)d(demonstrated)g (for)g(an)h(em)m(b)s(edding)d(in)150 2732 y(whic)m(h)39 b(all)h(\014v)m(e)f(unit)g(v)m(ectors)i(mak)m(e)e(a)g(star)h(lying)f (in)g(the)g(plane.)67 b(These)39 b(v)m(ectors)i(de\014ne)d(the)i(\014v) m(e)150 2845 y(v)m(ertices)d(of)f(a)g(symmetric)e(p)s(en)m(tagon.)57 b(Eac)m(h)36 b(angle)h(of)f(the)g(p)s(en)m(tagon)g(is)f(3)p Fm(\031)s(=)p Fn(5)h Fm(>)e(\031)s(=)p Fn(2,)k(and)d(hence)150 2958 y(this)30 b(is)h(not)f(an)h(em)m(b)s(edding)d(that)j(satis\014es)g (the)g(triangle)g(constrain)m(ts.)291 3071 y(The)h(ab)s(o)m(v)m(e)h (argumen)m(t)e(can)h(b)s(e)g(generalized)h(in)f(t)m(w)m(o)h(resp)s (ects,)f(sho)m(wn)g(in)f(the)h(next)g(t)m(w)m(o)i(prop)s(o-)150 3184 y(sitions.)150 3396 y Fg(Prop)s(osition)g(12)46 b Ff(F)-7 b(or)37 b(arbitr)-5 b(ary)39 b(gr)-5 b(aphs,)39 b(if)d(the)g(optimal)i(ve)-5 b(ctor)37 b(solution)g(to)g(the)g(SDP)e (with)i(tri-)150 3509 y(angle)c(c)-5 b(onstr)g(aints)35 b(lies)e(in)f(the)h(plane,)h(then)f(sdp)26 b Fn(=)f Ff(opt)q(.)291 3722 y Fg(Pro)s(of:)43 b Fn(Consider)30 b(an)m(y)i(three)f(unit)g(v)m (ectors)i(in)e(the)g(plane.)43 b(Then)30 b(either)i(at)g(least)g(t)m(w) m(o)h(of)e(them)150 3835 y(lie)h(on)f(the)h(same)f(line)h(\(though)f(p) s(ossibly)g(in)g(opp)s(osite)g(directions\),)i(or)e(the)h(endp)s(oin)m (ts)f(of)g(the)h(three)150 3947 y(mak)m(e)j(a)g(triangle)g(in)g(whic)m (h)f(one)h(of)g(the)f(angles)i(is)e(obtuse,)i(or)f(w)m(e)g(can)g (negate)h(one)f(of)g(the)f(v)m(ectors)150 4060 y(and)29 b(get)h(a)g(triangle)h(with)e(an)g(obtuse)g(angle.)42 b(Hence)30 b(an)m(y)f(v)m(ector)i(solution)f(in)f(the)h(plane)f(that)h (ob)s(eys)150 4173 y(the)38 b(acute)i(angle)f(rule)f(m)m(ust)f(ha)m(v)m (e)j(all)f(v)m(ectors)h(lie)f(on)f(just)g(t)m(w)m(o)h(lines)g(in)m (tersecting)h(in)e(the)g(origin)150 4286 y(\(though)28 b(on)f(eac)m(h)i(line)f(the)g(v)m(ectors)h(can)f(p)s(oin)m(t)g(in)f (either)i(direction\).)40 b(It)28 b(can)g(b)s(e)f(sho)m(wn)g(that)i(in) e(this)150 4399 y(case)h(the)f(ob)5 b(jectiv)m(e)29 b(function)d(of)h (the)g(SDP)f(is)h(linear)g(in)g(the)g(cosine)g(of)g(angle)h(b)s(et)m(w) m(een)f(the)g(t)m(w)m(o)h(lines,)150 4512 y(implying)33 b(that)i(the)f(maxim)m(um)e(is)i(ac)m(hiev)m(ed)i(when)d(the)i(angle)g (is)f(either)h(0)g(or)f Fm(\031)s Fn(.)52 b(But)35 b(in)f(this)g(case) 150 4625 y(w)m(e)j(ha)m(v)m(e)h(a)g(one-dimensional)e Fl(\006)p Fn(1)h(solution,)j(or)c(in)h(other)g(w)m(ords,)h(a)g(true)e (cut)h(in)g(the)g(graph.)59 b(The)150 4738 y(n)m(um)m(b)s(er)28 b(of)j(edges)g(cut)f(is)h(exactly)h(equal)f(to)g(the)f(ob)5 b(jectiv)m(e)33 b(function)d(of)g(the)h(SDP)-8 b(.)106 b Fa(2)150 4950 y Fg(Prop)s(osition)34 b(13)46 b Ff(With)33 b(the)g(triangle)g(c)-5 b(onstr)g(aints,)36 b(for)d(every)f(cycle)h (sdp)26 b Fn(=)f Ff(opt)q(.)1905 5247 y Fn(15)p eop %%Page: 16 16 16 15 bop 291 -58 a Fg(Pro)s(of:)41 b Fn(Recall)32 b(that)f(sdp)25 b Fl(\025)g Fn(opt.)41 b(Hence)31 b(w)m(e)g(need)f(only)g(sho)m(w)g (that)h(sdp)25 b Fl(\024)f Fn(opt)q(.)291 55 y(Ev)m(en-length)39 b(cycles)g(are)f(bipartite)g(graphs.)62 b(F)-8 b(or)38 b(ev)m(ery)h(bipartite)f(graph)f(the)h(maxim)m(um)d(cut)150 168 y(includes)22 b(all)h(the)f(edges.)38 b(As)22 b(the)g(v)-5 b(alue)23 b(of)f(the)g(SDP)g(cannot)h(b)s(e)e(more)g(than)h(all)h(the)f (edges,)i(sdp)g(=)h(opt)q(.)150 281 y(\(Moreo)m(v)m(er,)31 b(in)d(the)g(optimal)g(v)m(ector)h(solution)g(all)f(edges)h(m)m(ust)e (mak)m(e)h(an)f(angle)i(of)f Fm(\031)s Fn(,)h(and)e(a)i(random)150 394 y(h)m(yp)s(erplane)g(cuts)i(them)e(with)h(probabilit)m(y)h(1.\))291 506 y(It)j(remains)f(to)i(consider)f(o)s(dd-length)g(cycles.)52 b(F)-8 b(or)35 b(a)g Fm(k)s Fn(-cycle)h(with)d Fm(k)k Fn(o)s(dd,)e(the)f(maxim)m(um)d(cut)150 619 y(con)m(tains)37 b Fm(k)27 b Fl(\000)c Fn(1)36 b(edges.)57 b(Hence)36 b(w)m(e)g(need)f(to)h(sho)m(w)g(that)g(sdp)d Fl(\024)g Fm(k)27 b Fl(\000)c Fn(1.)57 b(F)-8 b(or)36 b(3-cycles)i (\(triangles\),)150 732 y(this)32 b(is)g(a)h(direct)f(consequence)h(of) f(constrain)m(t)h(3.)47 b(F)-8 b(or)33 b(o)s(dd)e(cycles)i(\()p Fm(v)2612 746 y Fj(1)2652 732 y Fm(;)15 b(:)g(:)g(:)h(v)2857 747 y Fk(k)2900 732 y Fn(\))32 b(of)h(length)f(at)h(least)g(5,)150 845 y(sum)40 b(up)h Fm(v)541 859 y Fj(1)581 845 y Fm(v)625 859 y Fk(j)690 845 y Fn(+)27 b Fm(v)832 859 y Fk(j)869 845 y Fm(v)913 859 y Fk(j)t Fj(+1)1068 845 y Fn(+)h Fm(v)1211 859 y Fk(j)t Fj(+1)1337 845 y Fm(v)1381 859 y Fj(1)1466 845 y Fl(\025)45 b(\000)p Fn(1)d(\(constrain)m(t)i(3\))f(o)m(v)m(er)g (all)g(ev)m(en)g(2)j Fl(\024)e Fm(j)51 b Fl(\024)45 b Fm(k)31 b Fl(\000)d Fn(1,)46 b(and)150 958 y Fl(\000)p Fm(v)265 972 y Fj(1)304 958 y Fm(v)348 972 y Fk(j)411 958 y Fn(+)27 b Fm(v)553 972 y Fk(j)589 958 y Fm(v)633 972 y Fk(j)t Fj(+1)787 958 y Fl(\000)f Fm(v)928 972 y Fk(j)t Fj(+1)1055 958 y Fm(v)1099 972 y Fj(1)1179 958 y Fl(\025)41 b(\000)p Fn(1)f(\(constrain)m(t)i(4\))e(o)m(v)m(er)h(all)g (o)s(dd)e(3)j Fl(\024)f Fm(j)46 b Fl(\024)41 b Fm(k)30 b Fl(\000)c Fn(2,)43 b(giving)e(\(after)150 1071 y(cancellations\))719 1007 y Fe(P)822 1071 y Fm(v)866 1085 y Fk(i)894 1071 y Fm(v)938 1085 y Fk(i)p Fj(+1)1082 1071 y Fl(\025)25 b(\000)p Fm(k)19 b Fn(+)d(2,)29 b(implying)e(that)i(the)g(v)-5 b(alue)29 b(of)f(the)h(SDP)f(is)g(at)h(most)f Fm(k)19 b Fl(\000)d Fn(1.)116 b Fa(2)291 1184 y Fn(In)28 b(fact,)i(with)e (triangle)i(constrain)m(ts)g(the)f(SDP)f(giv)m(es)i(optimal)f(v)-5 b(alue)29 b(for)f(all)i(planar)e(graphs)g(\(b)s(e-)150 1297 y(cause)e(the)f(triangle)h(constrain)m(ts)g(de\014ne)f(the)g(cut)g (p)s(olytop)s(e)g(for)g(planar)g(graphs,)g(and)g(more)f(generally)-8 b(,)150 1410 y(for)30 b(graphs)g(that)h(do)f(not)h(con)m(tain)g Fm(K)1465 1424 y Fj(5)1535 1410 y Fn(as)g(a)g(minor)e([2]\).)291 1523 y(There)44 b(is)f(no)h(kno)m(wn)g(conjecture)g(of)g(whic)m(h)g (graph)f(giv)m(es)i(the)f(w)m(orst)h(in)m(tegralit)m(y)h(ratio)f(with) 150 1636 y(triangle)29 b(constrain)m(ts.)40 b(In)27 b(Section)h(3.3)h (w)m(e)f(sho)m(w)m(ed)f(an)h(in)m(tegralit)m(y)i(ratio)e(of)g(at)g (most)f(roughly)g(0)p Fm(:)p Fn(891.)150 1748 y(Prior)j(to)h(that,)g (the)g(bad)f(examples)g(w)m(ere)h(only)f(kno)m(wn)g(to)h(ha)m(v)m(e)h (in)m(tegralit)m(y)h(ratio)e(at)g(most)f(roughly)150 1861 y(0)p Fm(:)p Fn(96,)i(that)f(is)g(ev)m(en)g(w)m(orse)f(than)h(the) f(hardness)f(of)i(appro)m(ximation)f(result)g(for)g(MAX)h(CUT.)291 1974 y(Determining)36 b(the)g(true)f(in)m(tegralit)m(y)k(ratio)e(for)e (the)h(SDP)g(with)f(triangle)i(constrain)m(ts)g(is)f(an)g(im-)150 2087 y(p)s(ortan)m(t)31 b(op)s(en)e(question.)150 2331 y Fb(4.2)112 b(Other)38 b(v)-6 b(alid)36 b(constrain)m(ts)150 2502 y Fn(W)-8 b(e)38 b(describ)s(e)f(a)h(pro)s(cedure)e(for)h (generating)i(v)-5 b(alid)37 b(constrain)m(ts)i(for)e(MAX)h(CUT.)f (This)f(pro)s(cedure)150 2615 y(generalizes)26 b(the)e(pro)s(cedure)f 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3417 y Fm(w)1803 3431 y Fk(ij)1873 3356 y Fn(1)21 b Fl(\000)f Fm(v)2074 3370 y Fk(i)2102 3356 y Fm(v)2146 3370 y Fk(j)p 1873 3396 310 4 v 2005 3479 a Fn(2)2218 3417 y Fl(\024)25 b Fm(a)1272 b Fn(\(6\))150 3704 y(This)26 b(quadratic)h(constrain)m(t)h(of)e(the)h(geometric)h (program)d(for)h(MAX)i(CUT)e(is)g(linear)h(in)g(the)f(v)-5 b(ariables)150 3817 y Fm(y)195 3831 y Fk(ij)283 3817 y Fn(of)28 b(the)g(SDP)-8 b(.)28 b(W)-8 b(e)29 b(can)f(deriv)m(e)h (other)f(v)-5 b(alid)28 b(constrain)m(ts)h(b)m(y)f(using)f(the)h(fact)h (that)f(in)g(our)f(in)m(tended)150 3930 y(solutions,)35 b(the)f Fm(v)763 3944 y Fk(i)824 3930 y Fn(are)g Fl(\006)p Fn(1)g(v)-5 b(ariables.)50 b(This)33 b(means)g(that)h(negating)g(a)g(v) -5 b(ariable)35 b(still)f(de\014nes)f(a)g(cut,)150 4043 y(and)26 b(constrain)m(t)i(6)f(m)m(ust)e(still)i(hold.)40 b(Let)27 b Fm(b)1648 4057 y Fj(1)1687 4043 y Fm(;)15 b(:)g(:)g(:)i(;)e(b)1928 4058 y Fk(k)1997 4043 y Fn(b)s(e)26 b(an)h(arbitrary)f(sequence)h(of)g Fl(\006)p Fn(1)f(v)-5 b(alues.)40 b(Then)150 4156 y(w)m(e)31 b(can)g(generalize)h(constrain)m (t)f(6)g(to:)1507 4317 y Fe(X)1482 4500 y Fk(i;j)t Fi(2)p Fk(S)1667 4398 y Fm(w)1732 4412 y Fk(ij)1802 4336 y Fn(1)21 b Fl(\000)f Fm(b)1998 4350 y Fk(i)2026 4336 y Fm(b)2065 4350 y Fk(j)2101 4336 y Fm(v)2145 4350 y Fk(i)2174 4336 y Fm(v)2218 4350 y Fk(j)p 1802 4376 453 4 v 2005 4460 a Fn(2)2289 4398 y Fl(\024)25 b Fm(a)1201 b Fn(\(7\))150 4693 y Fg(De\014nition)33 b(1)46 b Ff(A)35 b Fn(v)-5 b(alid)33 b(linear)h(constrain)m(t)g(for)f(the)h(SDP)f(for)g(MAX)g(CUT) i Ff(is)g(any)h(c)-5 b(onstr)g(aint)38 b(that)150 4806 y(c)-5 b(an)33 b(b)-5 b(e)33 b(derive)-5 b(d)33 b(by)g(the)g(ab)-5 b(ove)33 b(pr)-5 b(o)g(c)g(ess.)44 b(F)-7 b(or)34 b(short,)g(we)e(c)-5 b(al)5 b(l)34 b(such)f(c)-5 b(onstr)g(aints)35 b Fn(v)-5 b(alid)30 b(constrain)m(ts)p Ff(.)1905 5247 y Fn(16)p eop %%Page: 17 17 17 16 bop 291 -58 a Fg(Remark:)37 b Fn(Our)22 b(de\014nition)h(of)g(v) -5 b(alid)24 b(constrain)m(ts)g(is)f(not)h(the)f(most)g(general)h(p)s 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b(alue)28 b(of)g(the)g(SDP)f (solution)h(is)g(reduced)f(b)m(y)g(adding)g(the)h(triangle)h(constrain) m(ts,)377 2992 y(it)35 b(mak)m(es)f(sense)h(to)g(\014nd)d(a)j(cut)g (using)f(the)g(GW)h(solution)g(rather)f(than)g(GW)p Fl(4)p Fn(,)i(as)f(GW)g(no)m(w)377 3105 y(has)29 b(an)f(in)m(tegralit)m(y)j (ratio)f(strictly)g(less)f(than)f(1,)i(and)e(hence)g(an)h(appro)m (ximation)f(ratio)i(strictly)377 3218 y(b)s(etter)h(than)f Fm(\013)p Fn(.)291 3417 y(Let)e(us)f(also)i(note)f(some)f(similarities) h(b)s(et)m(w)m(een)g(the)g(7-cycle)h(example)e(and)g(those)h(of)g([12)q (,)g(1].)40 b(The)150 3530 y(graphs)33 b(in)m(v)m(olv)m(ed)j(can)e(all) h(b)s(e)f(view)m(ed)g(as)h(con)m(taining)g(subsets)e(of)i(the)f(v)m (ertices)i(of)e(a)g Fm(d)p Fn(-dimensional)150 3643 y Fl(\006)p Fn(1)i(h)m(yp)s(ercub)s(e.)57 b(F)-8 b(or)38 b(the)e(7-cycle)j Fm(d)c Fn(=)g(7,)k(for)d(Karlo\013)7 b('s)37 b(graphs)f Fm(d)g Fn(can)h(b)s(e)f(arbitrarily)g(large)i(but) 150 3756 y(m)m(ust)d(b)s(e)g(ev)m(en,)j(and)d(for)h(the)g(graphs)f(of)h (Alon)g(and)f(Sudak)m(o)m(v)h Fm(d)h Fn(is)e(arbitrary)-8 b(.)58 b(An)35 b(edge)i(is)e(placed)150 3869 y(b)s(et)m(w)m(een)g(t)m (w)m(o)g(v)m(ertices)h(if)e(their)g(hamming)e(distance)j(is)f(exactly)i Fm(q)s Fn(,)f(where)e Fm(q)k Fn(is)d(a)h(carefully)f(c)m(hosen)150 3982 y(parameter.)39 b(F)-8 b(or)29 b(the)f(7-cycle,)i Fm(q)e Fn(=)d(6.)40 b(The)28 b(v)m(ector)h(solution)f(of)g(the)g(SDP)f (is)h(naturally)g(tak)m(en)h(as)f(the)150 4095 y(h)m(yp)s(ercub)s(e)i (em)m(b)s(edding.)43 b(By)32 b(Lemma)e(14)i(it)h(satis\014es)f(all)h(v) -5 b(alid)32 b(constrain)m(ts,)h(and)e(b)m(y)h(Corollary)g(15)150 4208 y(the)f(graph)e(con)m(tains)j(a)f(cut)f(whose)g(v)-5 b(alue)31 b(is)g(at)g(least)g(equal)g(to)g(that)g(of)g(this)f(SDP)g (solution.)291 4320 y(In)24 b(these)i(graphs,)f(all)h(edges)g(mak)m(e)e (angles)i(whose)f(inner)f(pro)s(duct)g(is)h(\()p Fm(d)9 b Fl(\000)g Fn(2)p Fm(q)s Fn(\))p Fm(=d)p Fn(.)41 b(The)24 b(parameter)150 4433 y Fm(q)j Fn(is)d(c)m(hosen)h(so)f(that)h(the)g (angle)g(implied)e(b)m(y)h(this)g(inner)g(pro)s(duct)f(is)h(as)g(close) i(as)e(p)s(ossible)g(to)h(the)g(w)m(orst)150 4546 y(case)e(angle)g(for) f(the)g(random)e(h)m(yp)s(erplane)i(rounding)e(tec)m(hnique.)39 b(The)21 b(main)g(tec)m(hnical)j(issue)e(in)m(v)m(olv)m(ed)150 4659 y(in)j(these)g(examples)f(is)h(sho)m(wing)g(that)h(the)f(h)m(yp)s (ercub)s(e)e(em)m(b)s(edding)g(indeed)i(maximizes)f(the)h(SDP)-8 b(.)25 b(F)-8 b(or)150 4772 y(the)27 b(graphs)f(of)h([12)q(,)f(1)q(])h (the)g(pro)s(of)e(is)i(quite)g(in)m(v)m(olv)m(ed.)41 b(T)-8 b(o)27 b(some)f(exten)m(t,)j(it)f(is)e(ev)m(en)i(coinciden)m (tal.)41 b(F)-8 b(or)150 4885 y(example,)25 b(in)e(the)h(graphs)f(of)h (Alon)g(and)f(Sudak)m(o)m(v,)j(if)e Fm(q)i Fn(is)e(c)m(hosen)g (carelessly)h(to)g(b)s(e)e(o)s(dd,)h(the)g(resulting)150 4998 y(graphs)36 b(are)g(bipartite,)j(and)d(the)g(true)g(optimal)g(cut) h(giv)m(es)g(a)g(one)g(dimensional)e(v)m(ector)j(solution)f(of)1905 5247 y(19)p eop %%Page: 20 20 20 19 bop 150 -58 a Fn(m)m(uc)m(h)35 b(b)s(etter)g(v)-5 b(alue)36 b(than)g(the)g(high)f(dimensional)g(h)m(yp)s(ercub)s(e)f(em)m (b)s(edding.)54 b(F)-8 b(or)37 b(the)e(7-cycle,)40 b(the)150 55 y(pro)s(of)35 b(of)h(the)g(optimalit)m(y)h(of)f(the)g(h)m(yp)s (ercub)s(e)e(em)m(b)s(edding)g(b)s(ecomes)i(straigh)m(tforw)m(ard,)i (due)d(to)i(the)150 168 y(use)30 b(of)h(the)f(triangle)i(constrain)m (ts.)150 411 y Fb(4.4)112 b(Rounding)38 b(with)e(arbitrary)h(h)m(yp)s (erplanes)150 583 y Fn(The)25 b(7-cycle)i(example)e(sho)m(ws)g(that)g (for)g(GW)p Fl(4)p Fn(,)i(exp)p Fm(=)p Fn(opt)f Fm(<)f Fn(0)p Fm(:)p Fn(8788.)41 b(Ho)m(w)m(ev)m(er,)29 b(this)c(example)g (lea)m(v)m(es)150 696 y(m)m(uc)m(h)31 b(to)i(b)s(e)e(desired.)45 b(It)32 b(addresses)g(only)g(the)g(exp)s(ected)g(size)h(of)f(the)g (cut,)h(but)e(not)i(the)f(actual)h(size)150 809 y(of)i(the)g(cut)g (found)e(b)m(y)i(the)g(algorithm.)54 b(In)34 b(fact,)j(it)f(can)f(b)s (e)f(sho)m(wn)g(that)h(with)g(high)f(probabilit)m(y)-8 b(,)38 b(a)150 921 y(random)29 b(h)m(yp)s(erplane)g(pro)s(duces)g(an)h (optimal)h(cut.)291 1034 y(Let)26 b Fm(p)495 1048 y Fk(i)548 1034 y Fn(b)s(e)e(the)h(probabilit)m(y)h(that)f(the)h(random)d(h)m(yp)s (erplane)h(pro)s(duces)g(a)h(cut)g(with)g Fm(i)g Fn(edges.)40 b(Then)150 1083 y Fe(P)253 1147 y Fm(i)22 b Fl(\001)g Fm(p)399 1161 y Fk(i)455 1147 y Fn(=)29 b(exp)f Fl(')h Fn(0)p Fm(:)p Fn(8788opt)j Fl(\025)c Fn(5)p Fm(:)p Fn(27.)49 b(Observ)m(e)32 b(that)h(ev)m(ery)g(cut)g(of)g(the)g(7-cycle)h(con)m (tains)g(an)e(ev)m(en)150 1260 y(n)m(um)m(b)s(er)38 b(of)j(edges.)72 b(Hence)1195 1196 y Fe(P)1282 1283 y Fk(i)1326 1260 y Fm(i)27 b Fl(\001)g Fm(p)1482 1274 y Fk(i)1553 1260 y Fl(\024)42 b Fn(6)p Fm(p)1757 1274 y Fj(6)1823 1260 y Fn(+)27 b(4\(1)h Fl(\000)f Fm(p)2218 1274 y Fj(6)2257 1260 y Fn(\))43 b(=)f(2)p Fm(p)2539 1274 y Fj(6)2606 1260 y Fn(+)27 b(4.)71 b(It)41 b(follo)m(ws)h(that)f Fm(p)3516 1274 y Fj(6)3555 1260 y Fn(,)j(the)150 1373 y(probabilit)m(y)36 b(of)f(pro)s(ducing)f(a)i(cut)g(with)f(six)g (edges,)j(is)d(at)h(least)h(0)p Fm(:)p Fn(63.)57 b(The)35 b(conclusion)h(is)f(that)h(for)150 1486 y(algorithm)28 b(GW)p Fl(4)p Fn(,)i(the)f(7-cycle)h(is)f(an)f(example)g(of)h(a)g (graph)f(for)g(whic)m(h)g(exp)p Fm(=)p Fn(opt)h(is)f(v)m(ery)h(close)h (to)f Fm(\013)p Fn(,)150 1599 y(but)23 b(nev)m(ertheless)i(the)f (algorithm)g(most)g(lik)m(ely)h(do)s(es)f(pro)s(duce)f(the)h(optimal)g (cut.)39 b(That)24 b(is,)h(alg)h(=)f(opt)q(.)150 1712 y(It)32 b(is)f(natural)h(to)g(ask)f(whether)g(a)h(similar)f(phenomenon) e(happ)s(ens)h(for)h(Karlo\013)7 b('s)32 b(example.)44 b(Indeed,)150 1825 y(Karlo\013)33 b(in)g(his)f(pap)s(er)f([12)r(])h(is) h(cautious)h(enough)e(to)i(w)m(arn)e(the)h(reader)f(that)i(his)e (results)g(apply)h(only)150 1938 y(to)e(the)g(ratio)g(exp)p Fm(=)p Fn(opt)q(,)f(but)g(not)h(to)g(the)f(ratio)i(alg)q Fm(=)p Fn(opt)f(with)f(the)h(indep)s(enden)m(t)e(rep)s(etitions)h(p)s (olicy)-8 b(.)291 2051 y(A)31 b(simple)f(w)m(a)m(y)i(around)e(this)g (problem)g(is)g(as)i(follo)m(ws.)43 b(T)-8 b(ak)m(e)32 b Fm(n)e Fn(disjoin)m(t)i(copies)f(of)g(7-cycles,)j(and)150 2164 y(consider)j(a)g(7)p Fm(n)f Fn(dimensional)g(v)m(ector)i(solution) f(where)f(eac)m(h)i(cycle)g(is)f(em)m(b)s(edded)d(in)i(7-co)s (ordinates)150 2276 y(as)f(describ)s(ed)f(ab)s(o)m(v)m(e)j(for)e(the)g (7-cycle,)k(and)34 b(the)i(subspaces)e(\(nonzero)i(co)s(ordinates\))g (of)f(eac)m(h)i(cycle)150 2389 y(are)31 b(orthogonal)h(\(disjoin)m (t\).)43 b(This)30 b(v)m(ector)i(solution)g(ob)s(eys)e(the)h(triangle)h (constrain)m(ts)g(\(and)e(all)i(v)-5 b(alid)150 2502 y(constrain)m(ts\),)32 b(b)m(y)e(Lemma)f(16.)291 2615 y(Consider)39 b(no)m(w)h(a)g(random)e(h)m(yp)s(erplane,)j(whic)m(h)f (is)g(de\014ned)e(b)m(y)i(its)g(normal:)58 b(a)41 b(v)m(ector)g Fm(r)h Fn(in)e(a)150 2728 y(random)28 b(direction.)41 b(The)29 b(pro)5 b(jections)30 b(of)f Fm(r)j Fn(on)e(eac)m(h)h(of)e (the)h Fm(n)f Fn(7-dimensional)g(subspaces)g(are)h(again)150 2841 y(in)20 b(random)f(directions,)24 b(and)c(moreo)m(v)m(er,)k(indep) s(enden)m(t.)36 b(F)-8 b(rom)20 b(eac)m(h)i(7-cycle)h(the)e(random)e(h) m(yp)s(erplane)150 2954 y(it)38 b(is)f(exp)s(ected)g(to)h(cut)f (roughly)g(5)p Fm(:)p Fn(27)h(edges,)i(and)c(the)h(actual)h(n)m(um)m(b) s(er)d(of)i(edges)h(cut)f(is)g(b)s(ounded)150 3067 y(b)s(et)m(w)m(een) 32 b(0)g(and)e(6.)44 b(The)31 b(exp)s(ected)h(n)m(um)m(b)s(er)d(of)i (edges)h(cut)g(from)d(all)k(cycles)f(is)f(roughly)g(5)p Fm(:)p Fn(27)23 b Fl(\001)e Fm(n)31 b Fn(\(b)m(y)150 3180 y(the)d(linearit)m(y)h(of)f(the)f(exp)s(ectation\),)k(and)c(the)g (probabilit)m(y)h(of)g(deviation)h(from)d(exp)s(ectation)j(b)m(y)e Fm(k)3619 3114 y Fl(p)p 3695 3114 55 4 v 66 x Fm(n)150 3293 y Fn(b)s(eha)m(v)m(es)38 b(lik)m(e)h Fm(c)709 3260 y Fk(k)748 3236 y Fc(2)825 3293 y Fn(for)e(some)h Fm(c)g(<)f Fn(1)h(\(from)f(standard)g(b)s(ounds)f(on)h(sums)f(of)i(indep)s(enden)m (t)f(random)150 3406 y(v)-5 b(ariables)31 b(with)f(b)s(ounded)e(v)-5 b(alues\).)291 3518 y(Applying)31 b(the)g(same)f(reasoning)i(to)f (Karlo\013)7 b('s)32 b(example)f(rather)f(than)h(the)g(7-cycle)i (example,)e(w)m(e)150 3631 y(can)g(sho)m(w)f(the)g(follo)m(wing:)150 3844 y Fg(Theorem)35 b(18)46 b Ff(F)-7 b(or)36 b(algorithm)h(GW)f(and)g (for)f(any)h(variant)g(of)g(it)f(that)h(uses)f(additional)j(valid)e(c) -5 b(on-)150 3957 y(str)g(aints)30 b(\(such)g(as)f(the)g(triangle)g (ine)-5 b(qualities\),)30 b(the)f(appr)-5 b(oximation)33 b(r)-5 b(atio)30 b(is)f(no)g(b)-5 b(etter)29 b(than)h Fm(\013)f Ff(when)150 4070 y(the)e(indep)-5 b(endent)28 b(r)-5 b(ep)g(etitions)28 b(r)-5 b(ounding)27 b(p)-5 b(olicy)28 b(is)e(use)-5 b(d.)40 b(In)27 b(mor)-5 b(e)27 b(te)-5 b(chnic)g(al)28 b(terms:)39 b(for)27 b(every)f Fm(\017)g(>)f Fn(0)150 4183 y Ff(ther)-5 b(e)39 b(is)f(a)g(c)-5 b(onstant)40 b Fn(0)35 b Fl(\024)f Fm(c)i(<)e Fn(1)p Ff(,)40 b(such)e(that)h(for)f(in\014nitely)h(many)f(values)h(of)f Fm(n)f Ff(ther)-5 b(e)39 b(ar)-5 b(e)39 b Fm(n)p Ff(-no)-5 b(de)150 4296 y(gr)g(aphs)35 b(for)e(which)g(al)5 b(l)33 b(but)g(a)g Fm(c)1237 4263 y Fk(n)1317 4296 y Ff(fr)-5 b(action)34 b(of)f(hyp)-5 b(erplanes)35 b(cut)d(less)h(than)h Fn(\()p Fm(\013)21 b Fn(+)f Fm(\017)p Fn(\))p Ff(opt)34 b(e)-5 b(dges.)291 4508 y Fg(Pro)s(of:)46 b Fn(Let)33 b Fm(k)i Fn(b)s(e)d(a)h(su\016cien)m(tly)g(large)h(constan)m(t)g(suc)m (h)e(that)h(there)f(is)h(a)g Fm(k)i Fn(v)m(ertex)f(graph)e(\(e.g.,)150 4621 y(from)d(Karlo\013)7 b('s)32 b(family)e(of)h(graphs\))g(for)f (whic)m(h)h(exp)p Fm(=)p Fn(opt)26 b Fl(\024)g Fm(\013)21 b Fn(+)f Fm(\017=)p Fn(2,)32 b(and)f(let)g Fm(m)g Fn(b)s(e)f(the)h(n)m (um)m(b)s(er)e(of)150 4734 y(edges)j(in)g(suc)m(h)f(a)h(graph.)44 b(Observ)m(e)32 b(that)g Fm(m)c(<)f(k)1887 4701 y Fj(2)1926 4734 y Fn(.)45 b(W)-8 b(e)33 b(call)g(this)f Fm(k)s Fn(-v)m(ertex)h (graph)e(the)h Ff(b)-5 b(asic)32 b Fn(graph.)150 4847 y(F)-8 b(or)40 b(a)f(su\016cien)m(tly)h(large)g Fm(n)e Fn(divisible)h(b)m(y)g Fm(k)s Fn(,)j(the)d(graph)f Fm(G)h Fn(is)g(comp)s(osed)f(of)h Fm(n=k)j Fn(disjoin)m(t)d(copies)150 4960 y(of)f(the)f(basic)h(graph.)61 b(T)-8 b(o)38 b(em)m(b)s(ed)e Fm(G)p Fn(,)j(em)m(b)s(ed)c(eac)m(h)k(cop)m(y)f(of)g(the)f(basic)h (graph)f(with)g(a)h(h)m(yp)s(ercub)s(e)1905 5247 y(20)p eop %%Page: 21 21 21 20 bop 150 -58 a Fn(em)m(b)s(edding)38 b(in)h(a)h(disjoin)m(t)g(set) h(of)f(co)s(ordinates.)69 b(Similarly)39 b(to)h(the)g(argumen)m(t)f (giv)m(en)i(for)f(disjoin)m(t)150 55 y(unions)33 b(of)h(7-cycles,)j (standard)c(b)s(ounds)f(on)h(large)i(deviations)g(for)f(sums)d(of)j(b)s (ounded)e(indep)s(enden)m(t)150 168 y(random)39 b(v)-5 b(ariables)42 b(sho)m(w)g(that)f(the)h(probabilit)m(y)g(that)g(the)f(n) m(um)m(b)s(er)e(of)j(edges)f(cut)h(b)m(y)f(a)h(random)150 281 y(h)m(yp)s(erplane)d(deviates)i(from)e(its)h(exp)s(ectation)i(b)m (y)e(more)f(than)g Fm(\017mn=)p Fn(2)p Fm(k)44 b Fn(is)c(at)h(most)e Fm(c)3289 248 y Fk(n)3336 281 y Fn(,)k(for)d(some)150 394 y Fm(c)26 b(<)f Fn(1)30 b(that)h(dep)s(ends)e(only)h(on)h Fm(k)i Fn(and)d Fm(\017)p Fn(.)291 506 y(W)-8 b(e)40 b(note)e(that)h(the)g(em)m(b)s(edding)d(describ)s(ed)h(ab)s(o)m(v)m(e)j (satis\014es)f(all)g(v)-5 b(alid)38 b(constrain)m(ts.)66 b(F)-8 b(or)39 b(eac)m(h)150 619 y(basic)31 b(graph,)g(this)f(follo)m (ws)i(from)d(Lemma)g(14,)j(and)e(for)h(the)g(union)f(of)g(all)i(basic)f (graphs)f(this)h(follo)m(ws)150 732 y(from)j(Lemma)f(16.)56 b(Moreo)m(v)m(er,)39 b(this)c(em)m(b)s(edding)f(ac)m(hiev)m(es)j(a)f (maxim)m(um)c(v)-5 b(alue)35 b(for)g(the)h(SDP)-8 b(.)35 b(The)150 845 y(ob)5 b(jectiv)m(e)30 b(function)e(for)g Fm(G)g Fn(is)g(the)h(sum)d(of)j(ob)5 b(jectiv)m(e)30 b(functions)e(for)g(eac)m(h)h(of)g(the)f(copies)i(of)e(the)h(basic)150 958 y(graphs.)39 b(An)m(y)28 b(solution)h(for)f Fm(G)g Fn(induces)f(solutions)i(for)e(the)i(basic)f(graphs)g(in)f(a)i(natural) f(w)m(a)m(y)-8 b(,)30 b(and)e(the)150 1071 y(v)-5 b(alues)31 b(of)f(these)h(solutions)g(is)f(b)s(ounded)e(b)m(y)j(the)f(optim)m(um)f (for)h(the)g(basic)h(graph.)116 b Fa(2)291 1184 y Fn(Theorem)27 b(18)j(sho)m(ws)e(that)h(there)g(are)g(graphs)f(and)f(solutions)i(to)h (the)e(SDP)h(suc)m(h)f(that)h(essen)m(tially)150 1297 y(ev)m(ery)h(h)m(yp)s(erplane)f(giv)m(es)i(a)e(cut)h(whose)f(v)-5 b(alue)30 b(is)g(nearly)f(as)h(bad)f(as)g Fm(\013)19 b Fl(\001)g Fn(opt.)40 b(Ho)m(w)m(ev)m(er,)32 b(ev)m(en)f(for)e(the)150 1410 y(graphs)j(and)g(em)m(b)s(eddings)e(considered)i(in)h(Theorem)e (18,)j(there)e(are)h(h)m(yp)s(erplanes)f(that)h(pro)s(duce)e(an)150 1523 y(optimal)36 b(cut.)58 b(\(F)-8 b(or)37 b(example,)g(a)f(h)m(yp)s (erplane)f(whose)h(normal)f(is)h(a)g(v)m(ector)i(with)e(v)-5 b(alue)36 b(1)g(at)h(ev)m(ery)150 1636 y Fm(k)s Fn(th)g(co)s(ordinate,) j(and)c(0)h(elsewhere.\))62 b(Keeping)37 b(the)g(same)f(graphs)g(as)i (those)f(of)g(Theorem)f(18)i(but)150 1748 y(c)m(hanging)i(the)g(em)m(b) s(edding,)g(w)m(e)g(can)g(deriv)m(e)g(examples)f(in)g(whic)m(h)g Ff(every)h Fn(h)m(yp)s(erplane)e(giv)m(es)j(a)f(cut)150 1861 y(with)30 b(v)-5 b(alue)31 b(roughly)f Fm(\013)21 b Fl(\001)f Fn(opt.)150 2074 y Fg(Theorem)35 b(19)46 b Ff(F)-7 b(or)35 b(algorithms)h(GW)e(and)h(GW)p Fl(4)p Ff(,)e(any)i(r)-5 b(ounding)35 b(te)-5 b(chnique)34 b(that)h(pr)-5 b(o)g(duc)g(es)36 b(a)e(cut)150 2187 y(induc)-5 b(e)g(d)35 b(by)f(a)h(hyp)-5 b(erplane)37 b(\(such)e(as)g(the)f(der)-5 b(andomize)g(d)38 b(version)d(of)f(algorithm)j(GW\))d(has)i(appr)-5 b(ox-)150 2300 y(imation)37 b(r)-5 b(atio)38 b(no)e(b)-5 b(etter)37 b(than)g Fm(\013)p Ff(.)52 b(In)36 b(mor)-5 b(e)37 b(te)-5 b(chnic)g(al)38 b(terms:)49 b(for)37 b(every)e Fm(\017)d(>)f Fn(0)36 b Ff(ther)-5 b(e)37 b(is)f(a)g(gr)-5 b(aph)150 2413 y(and)39 b(an)f(optimal)i(solution)g(to)e(the)g(SDP)g (for)g(which)h(every)f(hyp)-5 b(erplane)41 b(cuts)c(less)i(than)g Fn(\()p Fm(\013)25 b Fn(+)f Fm(\017)p Fn(\))p Ff(opt)150 2526 y(e)-5 b(dges.)291 2738 y Fg(Pro)s(of:)37 b Fn(The)20 b(graphs)f(considered)i(are)f(essen)m(tially)j(those)e(of)g(Theorem)e (18.)38 b(Let)21 b Fm(k)j Fn(b)s(e)19 b(a)i(su\016cien)m(tly)150 2851 y(large)29 b(constan)m(t)g(suc)m(h)f(that)g(there)g(is)g(a)g Fm(k)j Fn(v)m(ertex)e(graph)e(\(e.g.,)k(from)26 b(Karlo\013)7 b('s)28 b(family)g(of)g(graphs\))f(for)150 2964 y(whic)m(h)i(exp)p Fm(=)p Fn(opt)d Fl(\024)f Fm(\013)18 b Fn(+)g Fm(\017=)p Fn(3,)30 b(and)e(let)j Fm(m)d Fn(b)s(e)h(the)g(n)m(um)m(b)s(er)e(of)j (edges)f(in)g(suc)m(h)g(a)h(graph.)39 b(Observ)m(e)29 b(that)150 3077 y Fm(m)g(<)g(k)409 3044 y Fj(2)449 3077 y Fn(.)47 b(F)-8 b(or)34 b(a)f(su\016cien)m(tly)h(large)f Fm(n)g Fn(divisible)g(b)m(y)f Fm(k)s Fn(,)i Fm(G)e Fn(is)h(the)g(graph) f(comp)s(osed)g(of)h Fm(n=k)j Fn(disjoin)m(t)150 3190 y(copies)31 b(of)g(this)f(basic)h(graph.)291 3303 y(No)m(w)i(rather)e (than)h(em)m(b)s(edding)e(eac)m(h)j(basic)f(graph)g(on)f(a)i(disjoin)m (t)f(set)g(of)g(co)s(ordinates,)h(w)m(e)g(shall)150 3416 y(em)m(b)s(ed)d(the)i(graph)f(in)g(dimension)f Fm(d)p Fn(,)i(where)f Fm(d)h Fn(will)g(b)s(e)f(c)m(hosen)h(in)f(the)g(range)h Fm(k)f Fl(\024)c Fm(d)g(<)g(\017n=)p Fn(3)p Fm(k)s Fn(.)44 b(The)150 3528 y(em)m(b)s(edding)29 b(will)h(b)s(e)g(randomized.)40 b(In)30 b(other)g(w)m(ords,)h(w)m(e)g(shall)f(describ)s(e)g(a)h(large)h (family)d(of)i(optimal)150 3641 y(em)m(b)s(eddings,)37 b(and)g(will)g(sho)m(w)g(that)h(for)e(a)i(random)d(mem)m(b)s(er)f(of)k (this)f(family)-8 b(,)38 b(Theorem)e(19)i(holds.)150 3754 y(W)-8 b(e)32 b(will)e(not)h(exhibit)f(a)h(sp)s(eci\014c)f(em)m(b) s(edding,)f(as)i(this)f(is)h(not)f(required)g(b)m(y)g(the)h(Theorem.) 291 3867 y(F)-8 b(or)34 b(eac)m(h)g(of)g(the)f(copies)h(of)g(the)f (graph,)h(w)m(e)f(c)m(ho)s(ose)h(at)g(random)e(a)h Fm(k)s Fn(-dimensional)g(subspace)g(of)150 3980 y Fm(R)220 3947 y Fk(d)260 3980 y Fn(,)k(and)e(em)m(b)s(ed)f(this)i(cop)m(y)g(in)f (that)h(subspace)f(using)g(the)h(optimal)f(h)m(yp)s(ercub)s(e)f(em)m(b) s(edding.)54 b(The)150 4093 y(v)-5 b(alue)31 b(of)f Fm(d)h Fn(is)f(c)m(hosen)h(suc)m(h)f(that:)261 4306 y(1.)46 b(The)37 b(n)m(um)m(b)s(er)e(of)i(\\essen)m(tially)j(di\013eren)m(t")e (h)m(yp)s(erplanes)e(in)h Fm(R)2611 4273 y Fk(d)2688 4306 y Fn(is)g(small,)i(namely)-8 b(,)38 b(b)s(elo)m(w)g Fm(c)3703 4273 y Fk(n)377 4418 y Fn(where)d Fm(c)f(>)g Fn(1)i(is)f(some)g(constan)m(t)i(that)f(dep)s(ends)e(on)h Fm(d)p Fn(.)56 b(This)35 b(giv)m(es)i(an)e(upp)s(er)f(b)s(ound)f(on)i Fm(d)p Fn(.)377 4531 y(\()p Fm(d)d Fn(is)g(c)m(hosen)g(so)g(that)g(the) g(constan)m(t)h Fm(c)e Fn(is)h(smaller)f(than)g(the)h(in)m(v)m(erse)g (of)g(the)g(constan)m(t)h Fm(c)e Fn(from)377 4644 y(Theorem)e(18.\))261 4832 y(2.)46 b(The)29 b(triangle)i(constrain)m(ts)g(are)f(satis\014ed)g (with)f(high)h(probabilit)m(y)-8 b(.)41 b(This)29 b(giv)m(es)i(a)f(lo)m (w)m(er)h(b)s(ound)377 4945 y(on)37 b Fm(d)p Fn(.)59 b(It)36 b(is)h(needed)f(only)h(for)f(GW)p Fl(4)p Fn(,)j(but)c(not)i (for)f(GW.)i(Hence)f(for)f(GW)h(w)m(e)g(ma)m(y)f(simply)1905 5247 y(21)p eop %%Page: 22 22 22 21 bop 377 -58 a Fn(c)m(ho)s(ose)39 b Fm(d)g Fn(=)f Fm(k)s Fn(,)i(meaning)e(that)h(all)g(copies)g(are)f(em)m(b)s(edded)e (in)i(the)g(same)g(subspace,)i(but)d(at)377 55 y(random)29 b(orien)m(tations.)291 267 y(W)-8 b(e)35 b(no)m(w)e(upp)s(er)f(b)s (ound)g(the)h(n)m(um)m(b)s(er)f(of)h(essen)m(tially)j(di\013eren)m(t)e (h)m(yp)s(erplanes)e(in)i Fm(R)3278 234 y Fk(d)3318 267 y Fn(.)50 b(A)34 b(h)m(yp)s(er-)150 380 y(plane)28 b(is)h(de\014ned)e (b)m(y)h(its)h(normal.)39 b(Pic)m(k)29 b(a)g(small)f(v)-5 b(alue)29 b(of)f Fm(\015)i(>)25 b Fn(0)k(\(e.g.,)i Fm(\015)f Fn(=)25 b Fm(\017=d)p Fn(\).)41 b(W)-8 b(e)30 b(sa)m(y)f(that)g(t)m(w)m (o)150 493 y(h)m(yp)s(erplanes)23 b(are)i Ff(similar)h Fn(if)e(their)h(normals)e(mak)m(e)i(an)f(angle)i(of)e(at)h(most)f Fm(\015)5 b Fn(,)26 b(and)e Ff(di\013er)-5 b(ent)25 b Fn(otherwise.)150 606 y(Pic)m(king)i(normals)e(greedily)i(suc)m(h)f (that)h(no)f(t)m(w)m(o)i(are)e(of)h(distance)f(less)h(than)f Fm(\015)31 b Fn(from)25 b(eac)m(h)i(other,)h(there)150 719 y(are)39 b(only)g(\()p Fm(O)s Fn(\(1\))p Fm(=\015)5 b Fn(\))872 686 y Fk(d)953 719 y Fn(normals)38 b(in)g Fm(S)1481 686 y Fk(d)p Fi(\000)p Fj(1)1612 719 y Fn(.)65 b(This)38 b(de\014nes)g(a)h(family)f Fm(F)52 b Fn(of)38 b(\()p Fm(O)s Fn(\(1\))p Fm(=\015)5 b Fn(\))3170 686 y Fk(d)3252 719 y Fn(h)m(yp)s(erplanes,)150 832 y(suc)m(h)30 b(that)h(ev)m(ery)g(other)g(h)m(yp)s(erplane)e(is)i(similar)e(to)i (some)f(h)m(yp)s(erplane)g(in)g(the)g(family)-8 b(.)291 945 y(No)m(w)36 b(w)m(e)f(argue)g(that)g(the)g(cut)g(pro)s(duced)e(b)m (y)i(the)g(b)s(est)f(h)m(yp)s(erplane)g Fm(H)42 b Fn(is)34 b(appro)m(ximated)h(b)m(y)g(a)150 1058 y(cut)g(pro)s(duced)e(b)m(y)i(a) g(mem)m(b)s(er)e(of)i Fm(F)13 b Fn(.)54 b(W)-8 b(e)36 b(will)f(use)g(the)g(fact)h(that)f Fm(G)g Fn(is)g(nearly)g(regular,)h (and)e(that)150 1171 y(the)j(em)m(b)s(edding)f(on)h(the)g(sphere)f(of)h (the)h(basic)f(graphs)g(comp)s(osing)f Fm(G)g Fn(is)h(indep)s(enden)m (t.)60 b(F)-8 b(or)38 b(eac)m(h)150 1284 y(h)m(yp)s(erplane)32 b Fm(H)703 1251 y Fi(0)757 1284 y Fl(2)e Fm(F)13 b Fn(,)34 b(consider)g(a)f(slice)i(of)e(width)g Fm(\015)38 b Fn(around)33 b(it.)50 b(It)33 b(has)h(measure)e(at)i(most)f Fm(\017)g Fn(\(that)150 1396 y(dep)s(ends)e(on)i Fm(\015)5 b Fn(\).)50 b(Only)32 b(an)h Fm(\017)p Fn(-fraction)h(of)g(the)f(v)m(ertices)i(of)e Fm(G)g Fn(is)g(exp)s(ected)g(to)h(lie)g(there.)50 b(Moreo)m(v)m(er,)150 1509 y(with)28 b(probabilit)m(y)i Fm(c)858 1476 y Fi(\000)p Fk(\017n)1017 1509 y Fn(\(where)f Fm(c)d(>)e Fn(1)30 b(dep)s(ends)d(on)h Fm(k)s Fn(\))i(there)f(will)g(not)g(b)s(e)f(more)g (than)g(a)i(2)p Fm(\017)p Fn(-fraction)150 1622 y(of)g(the)h(v)m (ertices)h(there.)40 b(As)31 b(there)f(are)g(only)h(\()p Fm(O)s Fn(\(1\))p Fm(=\015)5 b Fn(\))2078 1589 y Fk(d)2151 1622 y Fn(h)m(yp)s(erplanes)29 b(in)h Fm(F)13 b Fn(,)30 b(the)g(same)g(holds)g(w.h.p.)150 1735 y(for)k(all)i(h)m(yp)s(erplanes) d(in)i Fm(F)47 b Fn(sim)m(ultaneously)-8 b(.)54 b(If)34 b Fm(H)1983 1702 y Fi(0)2041 1735 y Fn(is)g(closest)j(in)d Fm(F)48 b Fn(to)35 b Fm(H)7 b Fn(,)36 b(then)e(their)h(symmetric)150 1848 y(di\013erence)e(lies)g(in)f(this)g(slice.)47 b(Hence)33 b(the)g(cuts)f(that)h(they)f(induce)g(on)g Fm(G)g Fn(di\013er)g(b)m(y)g Fm(O)s Fn(\()p Fm(\017n)p Fn(\))g(v)m(ertices,)150 1961 y(and)e(b)m(y)g(near)g(regularit)m(y)-8 b(,)33 b(b)m(y)d(an)g Fm(O)s Fn(\()p Fm(\017)p Fn(\))h(fraction)g(of)g(the)f(edges.)291 2074 y(Cho)s(osing)d(the)g(parameters)f(suc)m(h)h(that)g(\()p Fm(O)s Fn(\(1\))p Fm(=\015)5 b Fn(\))2054 2041 y Fk(d)2097 2074 y Fm(c)2136 2041 y Fi(\000)p Fk(\017n)2292 2074 y Fm(<)25 b Fn(1)p Fm(=)p Fn(4)j(ensures)e(that)i(for)f(at)g(least)i(3) p Fm(=)p Fn(4)f(of)150 2187 y(the)h(random)f(em)m(b)s(eddings,)g(ev)m (ery)i(h)m(yp)s(erplane)e(cuts)h(no)g(more)g(that)g(\()p Fm(\013)19 b Fn(+)f Fm(\017)p Fn(\)opt)29 b(edges.)41 b(F)-8 b(or)30 b Fm(n)f Fn(large)150 2300 y(enough,)h(taking)i Fm(d)25 b Fn(=)g Fm(o)p Fn(\()p Fm(n=)15 b Fn(log)j Fm(n)p Fn(\))31 b(su\016ces.)291 2413 y(W)-8 b(e)29 b(no)m(w)e(sho)m(w)g(that) h(with)f(high)g(probabilit)m(y)g(the)h(triangle)g(constrain)m(ts)g(are) g(satis\014ed.)40 b(F)-8 b(or)28 b(three)150 2526 y(v)m(ertices)45 b(in)e(the)h(same)f(cop)m(y)-8 b(,)48 b(this)c(follo)m(ws)h(from)d(the) h(h)m(yp)s(ercub)s(e)f(em)m(b)s(edding.)79 b(F)-8 b(or)44 b(v)m(ertices)h(in)150 2638 y(di\013eren)m(t)36 b(copies,)h(this)e (follo)m(ws)i(from)c(the)j(fact)g(that)g(for)e(random)g(and)g(indep)s (enden)m(t)g(unit)h(v)m(ectors,)150 2751 y(the)h(exp)s(ected)h(v)-5 b(alue)36 b(of)g(the)h(absolute)f(v)-5 b(alue)37 b(of)f(their)g(inner)f (pro)s(duct)g(is)h(roughly)g(1)p Fm(=)3255 2674 y Fl(p)p 3331 2674 48 4 v 77 x Fm(d)q Fn(,)h(and)f(the)150 2864 y(probabilit)m(y)26 b(of)f(b)s(eing)f(larger)i(than)1407 2791 y Fe(p)p 1490 2791 133 4 v 73 x Fm(c=d)g Fn(is)f(exp)s(onen)m (tially)h(small)f(in)g Fm(c)p Fn(.)39 b(Hence)26 b(if)f Fm(d)g Fl(\035)g Fm(k)3306 2831 y Fj(2)3361 2864 y Fn(log)18 b Fm(n)24 b Fn(then)150 2977 y(with)32 b(o)m(v)m(erwhelming)g (probabilit)m(y)h(the)g(absolute)g(v)-5 b(alue)32 b(of)h(the)f(inner)g (pro)s(duct)f(of)h(an)m(y)h(t)m(w)m(o)g(v)m(ertices)150 3090 y(b)s(elonging)26 b(to)g(t)m(w)m(o)i(di\013eren)m(t)e(copies)g(is) g(b)s(elo)m(w)g(1)p Fm(=k)s Fn(.)40 b(A)26 b(triangle)h(constrain)m(t)g (in)m(v)m(olving)g(three)f(v)m(ertices)150 3203 y(from)37 b(three)i(di\013eren)m(t)f(copies)i(is)e(necessarily)h(satis\014ed.)65 b(F)-8 b(or)39 b(the)g(case)g(of)g(t)m(w)m(o)h(v)m(ertices)g(from)d (one)150 3316 y(cop)m(y)c(and)e(the)h(third)f(from)f(another)i(cop)m(y) -8 b(,)34 b(w)m(e)e(use)g(the)g(fact)h(that)f(if)g(the)g(t)m(w)m(o)h(v) m(ertices)g(of)f(the)h(same)150 3429 y(cop)m(y)h(lie)f(on)g(the)g(same) f(line)i(\(in)f(opp)s(osite)g(directions\),)h(the)f(triangle)h (inequalities)h(are)e(satis\014ed.)48 b(If)150 3542 y(these)33 b(v)m(ertices)h(lie)f(on)g(di\013eren)m(t)g(lines,)g(then)g(their)f (inner)g(pro)s(duct)f(is)i(at)g(least)h Fl(\000)p Fn(1)21 b(+)h(2)p Fm(=k)36 b Fn(\(this)d(is)g(a)150 3655 y(consequence)27 b(of)g(the)g(h)m(yp)s(ercub)s(e)e(em)m(b)s(edding\))g(and)h(the)h(t)m (w)m(o)h(other)f(inner)f(pro)s(ducts)f(in)m(v)m(olv)m(ed)j(in)f(the)150 3768 y(triangle)32 b(inequalit)m(y)f(ha)m(v)m(e)h(absolute)f(v)-5 b(alue)31 b(b)s(elo)m(w)f(1)p Fm(=k)s Fn(.)291 3880 y(By)h(c)m(ho)s (osing)g Fm(d)g Fn(suc)m(h)f(that)h Fm(k)1333 3847 y Fj(2)1387 3880 y Fn(log)18 b Fm(n)25 b Fl(\034)g Fm(d)h Fl(\034)f Fm(n=k)33 b Fn(our)d(theorem)g(is)g(pro)m(v)m(ed.)291 3993 y(W)-8 b(e)27 b(note)g(that)g(w)m(e)f(can)h(accommo)s(date)f(v)-5 b(alid)26 b(constrain)m(ts)h(other)f(than)g(the)g(triangle)i(constrain) m(ts,)150 4106 y(b)m(y)e(giving)i(a)f(higher)f(lo)m(w)m(er)i(b)s(ound)c (on)j Fm(d)p Fn(.)39 b(Ho)m(w)m(ev)m(er,)30 b(if)c(the)h(n)m(um)m(b)s (er)d(of)j(terms)f(in)g(a)h(constrain)m(t)g(gro)m(ws)150 4219 y(as)k(a)f(function)g(of)h Fm(n)p Fn(,)f(the)h(lo)m(w)m(er)g(b)s (ound)e(on)h Fm(d)g Fn(migh)m(t)h(o)m(v)m(ertak)m(e)i(the)e(upp)s(er)d (b)s(ound.)114 b Fa(2)150 4463 y Fb(4.5)e(Making)38 b(the)f(graph)i (connected)150 4634 y Fn(In)34 b(Theorem)g(19)i(w)m(e)f(sho)m(w)g(that) g(for)g(ev)m(ery)h Fm(\017)d(>)f Fn(0)k(there)f(is)g(a)g(graph)g(and)f (an)h(optimal)g(solution)g(to)150 4747 y(the)d(SDP)g(for)f(whic)m(h)h (ev)m(ery)g(h)m(yp)s(erplane)f(cuts)h(less)g(than)g(\()p Fm(\013)22 b Fn(+)e Fm(\017)p Fn(\)opt)33 b(edges.)45 b(Ho)m(w)m(ev)m(er,)35 b(the)d(graphs)150 4860 y(pro)s(duced)k(in)h (the)h(pro)s(of)f(of)g(Theorem)g(19)h(are)g(not)g(connected.)63 b(This)37 b(is)g(an)h(undesirable)e(feature,)150 4973 y(b)s(ecause)44 b(the)f(\\standard")h(approac)m(h)g(to)g(handling)f (graphs)g(that)h(are)g(not)g(connected)h(is)e(\014rst)g(to)1905 5247 y(22)p eop %%Page: 23 23 23 22 bop 150 -58 a Fn(separate)38 b(them)d(in)m(to)j(connected)f(comp) s(onen)m(ts,)g(and)f(then)h(run)e(the)h(MAX)i(CUT)d(algorithm)i (\(e.g.,)150 55 y(the)j(GW)g(algorithm\))g(on)g(eac)m(h)h(comp)s(onen)m (t)e(separately)-8 b(.)70 b(The)39 b(cuts)h(found)e(for)i(eac)m(h)h (comp)s(onen)m(t)150 168 y(can)34 b(easily)h(b)s(e)e(joined)h(in)m(to)h (one)f(global)h(cut.)52 b(Hence)35 b(inputs)e(to)h(the)g(GW)h (algorithm)f(are)g(t)m(ypically)150 281 y(assumed)29 b(to)i(b)s(e)e(connected)j(graphs.)291 394 y(In)c(this)h(section)h(w)m (e)f(sho)m(w)f(ho)m(w)h(to)h(mo)s(dify)c(the)j(graphs)f(pro)s(duced)f (in)i(the)g(pro)s(of)f(of)h(Theorem)e(19)150 506 y(so)34 b(as)f(to)i(mak)m(e)e(them)f(connected.)51 b(Recall)35 b(that)f(these)g(graphs)f(con)m(tained)i Fm(n=k)h Fn(comp)s(onen)m(ts,) e(eac)m(h)150 619 y(of)h(size)i Fm(k)s Fn(,)f(where)f Fm(k)k Fn(can)c(b)s(e)g(assumed)e(to)j(b)s(e)f(arbitrarily)g(large.)57 b(The)35 b(most)f(natural)i(approac)m(h)f(to)150 732 y(making)h(these)i(graphs)e(connected)i(is)f(to)g(pic)m(k)h(one)f(v)m (ertex)h(in)f(ev)m(ery)g(connected)h(comp)s(onen)m(t)e(\(call)150 845 y(these)d(v)m(ertices)i Fm(w)782 859 y Fj(1)837 845 y Fm(:)15 b(:)g(:)h(w)1023 864 y Fk(n=k)1144 845 y Fn(\),)34 b(and)e(then)h(add)f(the)h(edges)h(\()p Fm(w)2311 859 y Fk(i)2339 845 y Fm(;)15 b(w)2444 859 y Fk(i)p Fj(+1)2563 845 y Fn(\))33 b(for)g(1)d Fl(\024)f Fm(i)h(<)f(n=k)s Fn(.)49 b(The)32 b(graph)150 958 y(then)44 b(b)s(ecomes)g(connected.)83 b(The)44 b(n)m(um)m(b)s(er)e(of)i(edges)h(that)g(w)m(e)g(add)f(is)g (negligible)i(compared)d(to)150 1071 y(the)36 b(total)h(n)m(um)m(b)s (er)c(of)j(edges)g(in)f(the)h(graph)f(\(when)g Fm(k)j Fn(is)e(large)g(enough\),)i(and)c(hence)i(the)g(e\013ect)h(of)150 1184 y(this)d(mo)s(di\014cation)f(on)h(the)h(v)-5 b(alue)34 b(of)g(the)h(optimal)f(cut)g(is)g(negligible.)54 b(Lik)m(ewise,)36 b(the)e(e\013ect)i(of)e(this)150 1297 y(mo)s(di\014cation)i(on)h(the)g (v)-5 b(alue)38 b(of)f(the)g(SDP)g(\(either)h(GW)f(or)g(GW)p Fl(4)p Fn(\))h(is)f(negligible.)62 b(Ho)m(w)m(ev)m(er,)41 b(this)150 1410 y(approac)m(h)31 b(is)f(not)h(satisfactory)-8 b(,)33 b(b)s(ecause)d(it)h(migh)m(t)f(force)h(the)g(v)m(ector)h (con\014guration)f(of)f(the)h(optimal)150 1523 y(solution)e(to)h(the)f (SDP)f(to)h(b)s(e)f(di\013eren)m(t)i(than)e(that)h(assumed)e(in)i(the)f (pro)s(of)g(of)h(Theorem)f(19.)41 b(Indeed,)150 1636 y(the)35 b(angle)h(b)s(et)m(w)m(een)f(the)g(v)m(ector)h(asso)s(ciated)g (with)f Fm(w)2060 1650 y Fk(i)2123 1636 y Fn(and)f(the)h(v)m(ector)h (asso)s(ciated)g(with)f Fm(w)3456 1650 y Fk(i)p Fj(+1)3609 1636 y Fn(will)150 1748 y(b)s(e)30 b(exactly)i Fm(\031)s Fn(,)e(rather)g(than)g(random.)39 b(The)30 b(c)m(hange)h(in)f(the)h(v)m (ector)g(con\014guration)g(ma)m(y)f(lead)h(to)g(the)150 1861 y(situation)g(in)f(whic)m(h)g(a)h(h)m(yp)s(erplane)f(pro)s(duces)f (an)h(optimal,)h(or)f(near)g(optimal,)h(cut.)291 1974 y(T)-8 b(o)39 b(o)m(v)m(ercome)g(this)f(di\016cult)m(y)-8 b(,)41 b(rather)c(than)h(connecting)h Fm(w)2455 1988 y Fk(i)2521 1974 y Fn(and)f Fm(w)2771 1988 y Fk(i)p Fj(+1)2927 1974 y Fn(b)m(y)g(a)g(single)h(edge,)h(w)m(e)150 2087 y(connect)h(them)e(b)m(y)g(a)h(small)g(subgraph)e(that)i(w)m(e)h(call)g (a)f(\\connecting)h(gadget".)71 b(The)40 b(connection)150 2200 y(is)d(done)h(b)m(y)f(iden)m(tifying)h Fm(w)1130 2214 y Fk(i)1196 2200 y Fn(with)f(one)h(sp)s(ecially)g(c)m(hosen)g(v)m (ertex)h(of)e(the)h(connecting)g(gadget,)k(and)150 2313 y(iden)m(tifying)c Fm(w)674 2327 y Fk(i)p Fj(+1)829 2313 y Fn(with)f(another)g(sp)s(ecially)h(c)m(hosen)g(v)m(ertex)g(of)f(the)h (connecting)g(gadget.)63 b(The)36 b(k)m(ey)150 2426 y(prop)s(ert)m(y)30 b(of)g(a)h(connecting)h(gadget)f(is)g(summarized)d(in)i(the)g(follo)m (wing)i(de\014nition.)150 2617 y Fg(De\014nition)h(3)46 b Ff(A)30 b(c)-5 b(onne)g(cte)g(d)32 b(gr)-5 b(aph)33 b Fm(H)k Ff(is)31 b(a)g Fn(connecting)e(gadget)j Ff(\(for)g(SDPs)e(for) h(MAX)e(CUT\))i(if)f(it)150 2730 y(has)35 b(the)g(fol)5 b(lowing)35 b(pr)-5 b(op)g(erty.)50 b(L)-5 b(et)34 b Fm(G)g Ff(b)-5 b(e)34 b(an)h(arbitr)-5 b(ary)36 b(gr)-5 b(aph.)49 b(L)-5 b(et)34 b Fm(G)2662 2744 y Fk(H)2763 2730 y Ff(b)-5 b(e)34 b(a)h(gr)-5 b(aph)36 b(obtaine)-5 b(d)36 b(fr)-5 b(om)150 2843 y Fm(G)30 b Ff(and)i Fm(H)38 b Ff(by)31 b(c)-5 b(onne)g(cting)32 b(two)g(vertic)-5 b(es)31 b(of)h Fm(G)e Ff(via)h Fm(H)7 b Ff(,)32 b(as)f(explaine)-5 b(d)33 b(ab)-5 b(ove.)42 b(Consider)32 b(an)g(arbitr)-5 b(ary)150 2956 y(optimal)36 b(ve)-5 b(ctor)36 b(c)-5 b(on\014gur)g(ation)36 b(for)f(the)g(GW)g(\(or)g(GW)p Fl(4)p Ff(\))g(SDP)f(on)g Fm(G)p Ff(.)47 b(Then)35 b(ther)-5 b(e)36 b(is)e(an)h(optimal)150 3069 y(ve)-5 b(ctor)40 b(c)-5 b(on\014gur)g(ation)42 b(for)e(the)g(GW)g(\(or)g(GW)p Fl(4)p Ff(\))g(SDP)f(on)h Fm(G)2398 3083 y Fk(H)2504 3069 y Ff(that)h(is)e(identic)-5 b(al)41 b(to)f(the)g(original)150 3182 y(c)-5 b(on\014gur)g(ation)35 b(on)e(al)5 b(l)33 b(original)h(vertic)-5 b(es)32 b(of)h Fm(G)p Ff(.)291 3373 y Fn(W)-8 b(e)35 b(no)m(w)g(describ)s(e)e(a)i(graph)e Fm(H)41 b Fn(that)35 b(has)f(12)h(v)m(ertices)h(and)d(is)i(a)f (connecting)h(gadget.)54 b(V)-8 b(ertices)150 3486 y Fl(f)p Fn(1)p Fm(;)15 b Fn(2)p Fm(;)g Fn(3)p Fl(g)46 b Fn(and)c(v)m(ertices)i Fl(f)p Fn(4)p Fm(;)15 b Fn(5)p Fm(;)g Fn(6)p Fl(g)45 b Fn(mak)m(e)e(a)g(complete)g(bipartite)g(graph)f Fm(K)2848 3500 y Fj(3)p Fk(;)p Fj(3)2943 3486 y Fn(.)77 b(Lik)m(ewise,)47 b(v)m(ertices)150 3599 y Fl(f)p Fn(7)p Fm(;)15 b Fn(8)p Fm(;)g Fn(9)p Fl(g)34 b Fn(and)c(v)m(ertices)i Fl(f)p Fn(10)p Fm(;)15 b Fn(11)p Fm(;)g Fn(12)p Fl(g)35 b Fn(mak)m(e)c(a)g(complete)h(bipartite)f(graph.)41 b(In)30 b(addition)h(include)g(the)150 3712 y(four)f(edges)h(\(1)p Fm(;)15 b Fn(7\),)32 b(\(1)p Fm(;)15 b Fn(10\))33 b(\(4)p Fm(;)15 b Fn(7\),)33 b(\(4)p Fm(;)15 b Fn(10\).)43 b(This)29 b(completes)i(the)f(description)h(of)f Fm(H)7 b Fn(.)291 3825 y(The)23 b(graph)h Fm(H)30 b Fn(has)24 b(9)7 b(+)g(9)g(+)g(4)26 b(=)f(22)g(edges.)39 b(It)24 b(has)g(a)g(cut)g(con)m(taining)h(20)g (edges.)39 b(In)23 b(fact,)k(there)d(are)150 3938 y(t)m(w)m(o)31 b(suc)m(h)e(cuts.)41 b(One)29 b(of)g(them)g(is)g Fl(f)p Fn(1)p Fm(;)15 b Fn(2)p Fm(;)g Fn(3)p Fm(;)g Fn(7)p Fm(;)g Fn(8)p Fm(;)g Fn(9)p Fl(g)35 b Fn(and)29 b(the)h(other)g(is)f Fl(f)p Fn(1)p Fm(;)15 b Fn(2)p Fm(;)g Fn(3)p Fm(;)g Fn(10)p Fm(;)g Fn(11)p Fm(;)g Fn(12)q Fl(g)p Fn(.)47 b(Lik)m(ewise,)150 4051 y(the)31 b(follo)m(wing)g(also)h(holds:)150 4242 y Fg(Lemma)j(20)46 b Ff(F)-7 b(or)33 b(the)g(gr)-5 b(aph)35 b Fm(H)k Ff(describ)-5 b(e)g(d)34 b(ab)-5 b(ove,)33 b(the)g(optimal)i (value)d(of)h(the)g(GW)g(SDP)f(is)h(20.)291 4433 y Fg(Pro)s(of:)41 b Fn(The)28 b(optimal)g(v)-5 b(alue)29 b(of)g(the)f(GW)i(SDP)e(is)g(at) i(least)g(20,)g(as)e Fm(H)36 b Fn(has)28 b(a)h(cut)g(with)f(20)h (edges.)150 4546 y(W)-8 b(e)36 b(will)g(no)m(w)f(sho)m(w)g(that)h(the)f (optimal)h(v)-5 b(alue)35 b(of)h(the)f(SDP)g(is)g(at)h(most)f(20.)56 b(T)-8 b(o)35 b(do)g(this,)i(w)m(e)f(shall)150 4659 y(use)g(the)h(fact) g(that)g(a)g(semide\014nite)f(maximization)g(problem)f(has)h(a)h(dual)f (minimization)f(problem,)150 4772 y(and)28 b(the)i(optimal)f(v)-5 b(alue)29 b(of)g(the)h(minimization)e(problem)f(is)i(at)h(least)g(as)g (large)g(as)f(the)g(optimal)g(v)-5 b(alue)150 4885 y(of)36 b(the)f(maximization)g(problem.)54 b(Hence,)38 b(to)e(b)s(ound)e(from)f (ab)s(o)m(v)m(e)k(the)f(v)-5 b(alue)35 b(of)h(the)g(GW)g(SDP)-8 b(,)35 b(it)150 4998 y(su\016ces)30 b(to)h(sho)m(w)f(a)h(feasible)g (solution)g(to)g(the)g(dual)f(minimization)g(problem)e(of)j(v)-5 b(alue)31 b(20.)1905 5247 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1311 y Fj(1)2911 1297 y Fn(\))13 b(+)g Fm(\025)3096 1311 y Fk(n)3144 1297 y Fn(\()p Fm(A)3247 1311 y Fj(2)3287 1297 y Fn(\).)40 b(It)27 b(follo)m(ws)150 1410 y(that)i(for)e(our)h(matrix)f Fm(A)p Fn(,)i Fm(\025)1109 1424 y Fj(10)1209 1410 y Fl(\025)c(\000)p Fn(2.)40 b(It)28 b(can)g(b)s(e)g(seen)g(that)h Fm(A)f Fn(has)f(t)m(w)m(o)j(eigen)m(v)-5 b(alues)30 b(of)e(v)-5 b(alue)28 b Fl(\000)p Fn(3,)h(b)m(y)150 1523 y(considering)i(the)f(t)m (w)m(o)i(linearly)f(indep)s(enden)m(t)e(eigen)m(v)m(ectors)995 1727 y(\(+1)p Fm(;)15 b Fn(+1)p Fm(;)g Fn(+1)p Fm(;)g Fl(\000)p Fn(1)p Fm(;)g Fl(\000)p Fn(1)p Fm(;)g Fl(\000)p Fn(1)p Fm(;)g Fl(\000)p Fn(1)p Fm(;)g Fl(\000)p Fn(1)p Fm(;)g Fl(\000)p Fn(1)p Fm(;)g Fn(+1)p Fm(;)g Fn(+1)q Fm(;)g Fn(+1\))150 1931 y(and)983 2044 y(\(+1)p Fm(;)g Fn(+1)p Fm(;)g Fn(+1)p Fm(;)g Fl(\000)p Fn(1)p Fm(;)g Fl(\000)p Fn(1)p Fm(;)g Fl(\000)p Fn(1)p Fm(;)g Fn(+1)p Fm(;)g Fn(+1)p Fm(;)g Fn(+1)p Fm(;)g Fl(\000)p Fn(1)p Fm(;)g Fl(\000)p Fn(1)p Fm(;)g Fl(\000)p Fn(1)q(\))p Fm(:)150 2211 y Fn(These)41 b(eigen)m(v)-5 b(alues)43 b(m)m(ust)c(corresp)s(ond)h(to)i Fm(\025)1789 2225 y Fj(11)1905 2211 y Fn(and)e Fm(\025)2145 2225 y Fj(12)2220 2211 y Fn(.)72 b(Hence)42 b(the)f(term)f Fm(n\025)3101 2225 y Fj(12)3176 2211 y Fm(=)p Fn(4)h(con)m(tributes)150 2324 y(12)21 b Fl(\001)g Fn(3)p Fm(=)p Fn(4)26 b(=)f(9,)31 b(b)s(ounding)e(the)h(v)-5 b(alue)31 b(of)g(the)f(SDP)g(b)m(y)h(11)21 b(+)f(9)25 b(=)g(20.)118 b Fa(2)291 2437 y Fn(T)-8 b(ak)m(e)28 b(arbitrarily)f(one)g(v)m(ertex)g(from)e(the)i(\014rst)f Fm(K)2000 2451 y Fj(3)p Fk(;)p Fj(3)2121 2437 y Fn(and)g(one)g(v)m (ertex)i(from)d(the)i(second)f Fm(K)3458 2451 y Fj(3)p Fk(;)p Fj(3)3553 2437 y Fn(.)39 b(F)-8 b(or)150 2550 y(concreteness)36 b(w)m(e)e(tak)m(e)j(v)m(ertices)f(6)f(and)e(7.)54 b(Consider)33 b(no)m(w)i(the)g(GW)g(SDP)f(on)g Fm(H)7 b Fn(.)53 b(W)-8 b(e)36 b(claim)e(that)150 2663 y(regardless)d(of)g (whic)m(h)g(unit)f(v)m(ectors)i(are)f(assigned)g(to)h(6)f(and)f(7,)i 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b(v)m(ertex)g(6)h(of)f Fm(H)40 b Fn(and)34 b Fm(w)1315 3354 y Fk(i)p Fj(+1)1467 3340 y Fn(with)f(v)m(ertex)i(7)f (of)g Fm(H)7 b Fn(.)51 b(No)m(w)35 b(an)m(y)f(v)m(ector)h(solution)g (to)f(the)g(GW)150 3453 y(SDP)27 b(that)g(is)h(optimal)f(for)f Fm(G)h Fn(can)g(b)s(e)g(completed)g(to)g(a)h(v)m(ector)g(solution)g (that)g(is)f(also)h(optimal)f(on)g(the)150 3566 y(union)34 b(of)h Fm(G)f Fn(and)g Fm(H)7 b Fn(.)53 b(Moreo)m(v)m(er,)38 b(this)d(completion)f(do)s(es)h(not)g(in)m(tro)s(duce)f(an)m(y)h(new)f (v)m(ectors)i(\(other)150 3679 y(than)27 b Fl(\000)p Fm(v)j Fn(and)d Fl(\000)p Fm(u)p Fn(\),)i(so)e(it)h(preserv)m(es)f(the) h(prop)s(ert)m(y)f(that)h(v)-5 b(alid)27 b(constrain)m(ts)i (\(including)e(the)h(triangle)150 3792 y(constrain)m(ts\))k(are)f(not)f (violated.)291 3905 y(It)25 b(follo)m(ws)h(that)f(the)g(graph)f Fm(H)32 b Fn(that)25 b(w)m(e)g(describ)s(ed)f(is)g(indeed)g(a)h (connecting)h(gadget.)41 b(Needless)25 b(to)150 4017 y(sa)m(y)-8 b(,)31 b(if)f(w)m(e)g(wish)e(to)j(connect)f(sev)m(eral)h (pairs)e(of)h(v)m(ertices)h(in)e(a)h(graph)f Fm(G)p Fn(,)h(w)m(e)g(can) f(use)h(sev)m(eral)h(disjoin)m(t)150 4130 y(copies)f(of)f(the)g(graph)g Fm(H)35 b Fn(for)29 b(this)g(purp)s(ose.)39 b(\(Ho)m(w)m(ev)m(er,)32 b(if)c(a)i(v)m(ertex)g Fm(v)i Fn(of)d Fm(G)g Fn(is)g(a)g(mem)m(b)s(er)d (of)k(sev)m(eral)150 4243 y(suc)m(h)g(pairs,)h(then)g(the)f(resp)s (ectiv)m(e)i(connecting)g(gadgets)g(will)f(ev)m(en)m(tually)i(also)f (share)e(a)h(v)m(ertex)h({)f(via)150 4356 y(the)g(pro)s(cess)e(of)i (iden)m(tifying)g(a)g(v)m(ertex)g(from)e(the)i(connecting)g(gadget)h (with)e Fm(v)s Fn(.\))291 4469 y(F)-8 b(rom)28 b(the)g(ab)s(o)m(v)m(e)i (discussion)e(\(including)g(the)g(fact)h(that)g(the)g(size)g Fm(k)i Fn(of)e(a)f(connected)i(comp)s(onen)m(t)150 4582 y(of)k(the)g(original)g(graph)f Fm(G)g Fn(can)h(b)s(e)f(arbitrarily)h (large,)i(whereas)d(the)h(n)m(um)m(b)s(er)d(of)j(v)m(ertices)h(in)e Fm(H)41 b Fn(is)34 b(a)150 4695 y(\014xed)c(constan)m(t\))i(it)f(follo) m(ws)g(that)g(Theorem)e(19)i(holds)f(also)i(for)e(connected)h(graphs.) 1905 5247 y(24)p eop %%Page: 25 25 25 24 bop 150 -58 a Fo(5)137 b(Conclusions)150 145 y Fn(The)30 b(w)m(ork)g(of)h(Go)s(emans)e(and)h(Williamson)h(sho)m(ws)f (that)h(for)f(algorithm)h(GW)1303 349 y Fm(\013)21 b Fl(\001)f Fn(sdp)k Fl(\024)h Fn(exp)g Fl(\024)g Fn(alg)i Fl(\024)e Fn(opt)g Fl(\024)g Fn(sdp)o Fm(:)150 553 y Fn(Hence)j(the)g(ratio)h(b)s(et)m(w)m(een)f(an)m(y)g(t)m(w)m(o)h(of)f (the)g(ab)s(o)m(v)m(e)g(quan)m(tities)h(is)f(at)g(least)h Fm(\013)d Fl(')f Fn(0)p Fm(:)p Fn(87856.)43 b(But)27 b(can)h(it)150 666 y(b)s(e)21 b(as)h(lo)m(w)h(as)f Fm(\013)h Fn(\(or)f(arbitrarily)g(close)h(to)g(it\)?)38 b(Karlo\013)23 b(sho)m(w)m(ed)f(graphs)f(with)h(exp)o Fm(=)p Fn(opt)h(approac)m(hing)g Fm(\013)150 779 y Fn(from)g(ab)s(o)m(v)m(e.)41 b(In)24 b(this)g(w)m(ork)h(w)m(e)h(sho)m(w)f(similar)f(results)g(for)h(opt)p Fm(=)p Fn(sdp)g(and)f(alg)q Fm(=)p Fn(opt)q(.)39 b(The)24 b(discussion)g(in)150 892 y(the)j(b)s(eginning)e(of)i(Section)g(4.4)h (regarding)e(the)h(7-cycle)i(example)d(illustrates)h(that)g(the)g (ratio)g(exp)p Fm(=)p Fn(alg)150 1005 y(can)38 b(also)h(b)s(e)f (signi\014can)m(t,)j(and)c(ma)m(y)h(sometimes)f(b)s(e)g(the)i(reason)f (for)g(m)m(uc)m(h)f(of)h(the)g(ratio)h(exp)p Fm(=)p Fn(opt)150 1118 y(in)32 b(examples)g(suc)m(h)g(as)g(those)h(of)f(Karlo\013.)47 b(F)-8 b(urthermore,)32 b(in)m(terpreting)h(alg)g(as)g(the)f(result)h (of)f(taking)150 1231 y(the)d(cut)g(pro)s(duced)e(b)m(y)i(the)g(b)s (est)g(h)m(yp)s(erplane,)f(Karlo\013)7 b('s)30 b(example)e(sho)m(ws)h (a)g(ratio)h(of)f Fm(\013)h Fn(for)e(exp)p Fm(=)p Fn(alg)r(,)150 1344 y(with)38 b(equalit)m(y)h(b)s(et)m(w)m(een)g(alg)g(and)e(opt.)63 b(In)37 b(the)i(con)m(text)g(of)f(Karlo\013)7 b('s)39 b(example,)g(taking)g(the)f(b)s(est)150 1457 y(h)m(yp)s(erplane)31 b(is)g(a)h(plausible)f(p)s(ossibilit)m(y)-8 b(,)33 b(b)s(ecause)f(the)f (em)m(b)s(edding)f(is)h(in)h Fm(O)s Fn(\(log)17 b Fm(n)p 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(further)e(and)h(rule)h(out)g(a)150 3037 y(larger)31 b(class)g(of)g(rounding)e(tec)m(hniques.)41 b(Let)31 b(us)f(elab)s(orate)h(on)g(this.)291 3150 y(Using)44 b(SDP)f(w)m(e)h(get)g(an)g(em)m(b)s(edding)d(of)j(the)f(v)m(ertices)j (of)d(the)h(graph)f Fm(G)f Fn(on)i(a)g(sphere)e Fm(S)3594 3117 y Fk(d)p Fi(\000)p Fj(1)3725 3150 y Fn(.)150 3263 y(A)g(rounding)e(tec)m(hnique)j(uses)e(this)h(em)m(b)s(edding)e(to)i (pro)s(duce)f(a)h(cut)g(in)f Fm(G)p Fn(.)74 b(W)-8 b(e)43 b(call)g(a)f(rounding)150 3376 y(tec)m(hnique)j Ff(oblivious)g Fn(if)g(it)g(divides)f(the)h(sphere)f(in)m(to)h(t)m(w)m(o)h(parts)e (\(of)i(arbitrary)e(shap)s(e\))g(without)150 3489 y(lo)s(oking)33 b(at)f(the)g(em)m(b)s(edding)e(of)i(the)g(graph)f(on)h(the)g(sphere,)f (iden)m(ti\014es)h(these)h(parts)e(as)h(t)m(w)m(o)h(sides)f(of)150 3602 y(a)f(cut,)g(and)f(places)i(v)m(ertices)g(of)f Fm(G)f Fn(in)g(the)h(side)f(of)h(the)g(cut)g(that)g(corresp)s(onds)e(to)j(the) e(part)h(in)f(whic)m(h)150 3715 y(their)37 b(resp)s(ectiv)m(e)h(v)m (ectors)g(lie)g(in)e(the)i(solution)f(to)h(the)f(SDP)-8 b(.)37 b(The)f(random)f(h)m(yp)s(erplane)h(rounding)150 3828 y(tec)m(hnique)41 b(is)g(oblivious.)71 b(A)41 b(consequence)g(of)g (Corollary)g(6)g(is)g(that)g(for)f(ev)m(ery)i Fm(\017)g(>)f Fn(0,)j(when)c(the)150 3941 y(dimension)27 b Fm(d)j Fn(is)f(large)h (enough,)f(then)g(for)f(ev)m(ery)i Fm(\031)s(=)p Fn(2)c Fm(<)f(\014)31 b(<)25 b(\031)20 b Fl(\000)d Fm(\017)29 b Fn(the)g(follo)m(wing)i(holds:)39 b(if)29 b(a)g(graph)150 4053 y(con)m(taining)39 b(a)f(single)h(edge)f(is)g(em)m(b)s(edded)e(on) h(the)h(sphere)f Fm(S)2342 4021 y Fk(d)p Fi(\000)p Fj(1)2510 4053 y Fn(suc)m(h)h(that)g(the)g(endp)s(oin)m(ts)f(of)h(the)150 4166 y(edge)28 b(are)g(at)g(distance)h(b)s(et)m(w)m(een)f Fm(\014)k Fn(and)27 b Fm(\014)20 b Fn(+)14 b Fm(\017)27 b Fn(from)g(eac)m(h)h(other,)h(then)e(a)h(random)e(h)m(yp)s(erplane)g (is)i(the)150 4279 y(oblivious)k(rounding)f(tec)m(hnique)i(with)f Ff(highest)g Fn(probabilit)m(y)h(of)f(cutting)h(the)f(edge)h(\(up)e(to) i(lo)m(w)f(order)150 4392 y(terms\).)39 b(Using)26 b(this)f(observ)-5 b(ation,)28 b(it)f(can)f(b)s(e)f(sho)m(wn)g(that)i(Theorem)e(18)h (extends)g(to)h(ev)m(ery)f(oblivious)150 4505 y(rounding)j(tec)m (hnique.)291 4618 y(W)-8 b(e)33 b(no)m(w)f(discuss)g(some)f (nonoblivious)h(rounding)f(tec)m(hniques)i(that)f(w)m(ere)h(used)e(for) h(MAX)h(CUT)150 4731 y(or)d(similar)g(problems.)291 4844 y(F)-8 b(eige)25 b(and)e(Go)s(emans)f([6])h(suggested)h(to)g(use)e (rotations)j(in)d(cases)i(where)e(the)h(sphere)g(has)f(a)i(distinct)150 4957 y(\\north)32 b(p)s(ole")h(\(and)e(\\south)i(p)s(ole"\).)46 b(Rotating)34 b(eac)m(h)g(v)m(ector)f(somewhat)f(to)m(w)m(ards)h(its)f (nearest)h(p)s(ole)1905 5247 y(25)p eop %%Page: 26 26 26 25 bop 150 -58 a Fn(prior)29 b(to)h(cutting)h(the)f(sphere)f(with)g (a)h(random)e(h)m(yp)s(erplane)h(can)h(lead)g(to)g(impro)m(v)m(ed)g (solutions.)40 b(The)150 55 y(usefulness)23 b(of)h(this)h(approac)m(h)f (w)m(as)h(demonstrated)e(for)h(problems)e(suc)m(h)i(as)h(MAX)f(2SA)-8 b(T,)25 b(where)f(there)150 168 y(is)k(a)h(unique)e(additional)i(v)m (ector)h Fm(v)1348 182 y Fj(0)1415 168 y Fn(that)f(is)f(in)m(terpreted) h(as)f(the)g(+1)h(direction)g(and)e(can)i(serv)m(e)f(as)h(the)150 281 y(north)c(p)s(ole.)39 b(T)-8 b(rying)26 b(to)h(apply)e(a)h(similar) f(idea)i(to)f(MAX)h(CUT,)e(one)h(should)f(\014rst)g(c)m(ho)s(ose)i(a)f (direction)150 394 y(that)33 b(can)g(serv)m(e)g(as)g(the)f(north)g(p)s (ole)g(of)h(the)g(sphere.)46 b(The)32 b(pro)s(of)f(of)i(Theorem)e(19)i (can)g(b)s(e)f(extended)150 506 y(to)38 b(sho)m(w)g(that)g(no)f(c)m (hoice)j(of)d(north)g(p)s(ole)h(and)f(rotation)i(function)e(giv)m(es)i (an)e(appro)m(ximation)h(ratio)150 619 y(b)s(etter)27 b(than)h Fm(\013)p Fn(.)40 b(This)26 b(is)i(based)f(on)g(the)h(observ) -5 b(ation)28 b(that)g(c)m(ho)s(osing)g(a)g(north)e(p)s(ole)i(at)g (random)e(giv)m(es)150 732 y(an)k(oblivious)h(rounding)e(tec)m(hnique.) 291 845 y(Zwic)m(k)f([19)q(])h(has)e(used)g(a)h(notion)h(of)f Ff(outwar)-5 b(d)33 b(r)-5 b(otations)30 b Fn(for)e(sev)m(eral)h (problems.)38 b(F)-8 b(or)29 b(MAX)f(CUT,)150 958 y(Zwic)m(k)35 b(observ)m(es)f(that)h(there)g(are)g(t)m(w)m(o)h(\\bad")e(angles)h(for) g(whic)m(h)f(the)g(random)f(h)m(yp)s(erplane)g(fails)i(to)150 1071 y(giv)m(e)f(an)e(exp)s(ectation)i(ab)s(o)m(v)m(e)g Fm(\013)p Fn(.)47 b(One)32 b(is)h(the)f(angle)i Fm(\022)g Fn(men)m(tioned)e(in)g(Section)h(1.)48 b(The)32 b(other)g(is)h(the)150 1184 y(trivial)h(angle)f(0,)h(for)f(whic)m(h)f(the)h(con)m(tribution)g (to)g(the)g(v)-5 b(alue)33 b(of)g(the)g(geometric)h(program)d(is)i(0,)h (and)150 1297 y(so)39 b(is)f(the)h(con)m(tribution)g(to)g(the)f(cut)h (pro)s(duced)e(b)m(y)h(a)h(h)m(yp)s(erplane.)64 b(Hence)39 b(w)m(orst)g(case)g(instances)150 1410 y(for)34 b(the)h(GW)g(algorithm) f(ma)m(y)g(ha)m(v)m(e)h(an)g(arbitrary)f(mixture)f(of)h(b)s(oth)g(t)m (yp)s(es)g(of)h(angles)g(for)f(pairs)g(of)150 1523 y(v)m(ertices)29 b(connected)g(b)m(y)f(edges.)40 b(In)27 b(the)h(extreme)f(case,)j (where)d(all)i(angles)f(are)h(0)f(\(though)g(this)f(w)m(ould)150 1636 y(nev)m(er)h(b)s(e)f(the)h(optimal)g(geometric)h(em)m(b)s (edding\))d(it)j(is)f(clear)h(that)f(a)g(random)e(h)m(yp)s(erplane)h(w) m(ould)h(not)150 1748 y(cut)i(an)m(y)g(edge,)h(whereas)e(ignoring)h (the)g(geometric)h(em)m(b)s(edding)d(and)h(giving)i(the)e(v)m(ertices)j Fl(\006)p Fn(1)e(v)-5 b(alues)150 1861 y(indep)s(enden)m(tly)34 b(at)i(random)e(is)h(exp)s(ected)h(to)g(cut)g(roughly)f(half)g(the)h (edges.)56 b(This)35 b(latter)h(rounding)150 1974 y(tec)m(hnique)29 b(is)g(equiv)-5 b(alen)m(t)30 b(to)f(\014rst)f(em)m(b)s(edding)f(the)h (v)m(ertices)i(as)f(m)m(utually)f(orthogonal)i(unit)e(v)m(ectors,)150 2087 y(and)j(then)g(cutting)h(with)f(a)h(random)e(h)m(yp)s(erplane.)43 b Ff(Outwar)-5 b(d)35 b(r)-5 b(otation)34 b Fn(is)e(a)g(tec)m(hnique)g (of)f(a)m(v)m(eraging)150 2200 y(b)s(et)m(w)m(een)j(the)f(t)m(w)m(o)h (em)m(b)s(eddings:)45 b(the)33 b(optimal)g(geometric)h(em)m(b)s(edding) d(on)i(one)g(set)h(of)f(co)s(ordinates)150 2313 y(and)h(the)h(m)m (utually)g(orthogonal)h(em)m(b)s(edding)d(on)i(another)g(set)h(of)f(co) s(ordinates.)55 b(It)35 b(can)g(b)s(e)f(used)g(in)150 2426 y(order)e(to)i(obtain)f(appro)m(ximation)g(ratios)h(b)s(etter)f (than)g Fm(\013)g Fn(whenev)m(er)g(a)g(substan)m(tial)h(fraction)g(of)f (the)150 2539 y(edges)f(ha)m(v)m(e)g(angles)g(0,)f(sho)m(wing)h(that)f (essen)m(tially)i(the)e(only)g(case)h(when)e(the)i(geometric)g(em)m(b)s (edding)150 2652 y(\(p)s(erhaps\))21 b(fails)i(to)g(ha)m(v)m(e)g(in)m (tegralit)m(y)i(ratio)f(b)s(etter)e(than)g Fm(\013)h Fn(is)f(when)f(all)i(edges)g(ha)m(v)m(e)g(angle)h Fm(\022)s Fn(.)37 b(W)-8 b(e)23 b(note)150 2765 y(that)34 b(the)f(cuts)h (obtained)f(b)m(y)g(\014rst)g(applying)g(out)m(w)m(ard)g(rotations)i (are)f(not)f(oblivious)h(cuts,)g(b)s(ecause)150 2878 y(t)m(w)m(o)c(v)m(ertices)g(that)f(w)m(ere)g(mapp)s(ed)d(to)j(the)g (same)f(unit)g(v)m(ector)i(ma)m(y)e(end)g(up)g(at)h(di\013eren)m(t)g (sides)f(of)h(the)150 2990 y(cut.)53 b(Theorem)33 b(2)h(do)s(es)g(not)h (apply)f(in)g(this)g(case.)53 b(Nev)m(ertheless,)37 b(w)m(e)e(note)g (that)g(the)f(idea)h(of)g(outer)150 3103 y(rotation)28 b(do)s(es)e(not)g(seem)g(to)h(pro)m(vide)g(an)m(y)g(adv)-5 b(an)m(tage)28 b(when)d(the)i(em)m(b)s(edding)e(is)h(suc)m(h)g(that)h (all)g(edges)150 3216 y(mak)m(e)j(angles)i(larger)f(than)f Fm(\031)s(=)p Fn(2,)h(whic)m(h)f(is)h(the)f(case)i(of)e(greatest)j(in)m (terest.)291 3329 y(Giv)m(en)42 b(a)g(cut)f(in)g(a)h(graph,)h(a)f Ff(misplac)-5 b(e)g(d)43 b Fn(v)m(ertex)f(is)g(one)f(that)h(is)f(on)g (the)g(same)g(side)g(as)h(most)150 3442 y(of)36 b(its)f(neigh)m(b)s (ors.)56 b(The)35 b(n)m(um)m(b)s(er)e(of)i(edges)h(cut)g(can)g(b)s(e)e (increased)i(b)m(y)g(ha)m(ving)g(misplaced)e(v)m(ertices)150 3555 y(c)m(hange)e(sides.)41 b(This)29 b(lo)s(cal)j(heuristic)f(w)m(as) g(used)e(b)m(y)i(F)-8 b(eige,)33 b(Karpinski)c(and)h(Langb)s(erg)g([7)q (])h(on)f(top)h(of)150 3668 y(the)j(random)d(h)m(yp)s(erplane)i (rounding)f(tec)m(hnique)i(to)g(obtain)g(an)f(appro)m(ximation)g(ratio) i(signi\014can)m(tly)150 3781 y(b)s(etter)41 b(than)f Fm(\013)j Fl(')f Fn(0)p Fm(:)p Fn(87856)i(for)c(MAX)h(CUT)f(on)h (graphs)f(with)g(b)s(ounded)f(degree)i(\(for)g(graphs)f(of)150 3894 y(degree)29 b(at)g(most)f(3)h(the)f(appro)m(ximation)h(ratio)g(is) f(b)s(etter)h(than)f(0)p Fm(:)p Fn(92\).)42 b(The)28 b(b)s(est)g(h)m(yp)s(erplane)f(cannot)150 4007 y(repro)s(duce)32 b(results)h(whic)m(h)g(are)h(as)g(strong,)g(b)s(ecause)g(the)f(pro)s (of)g(tec)m(hnique)h(of)f(Theorem)g(19)h(applies)150 4120 y(also)d(to)g(graphs)f(of)h(v)m(ery)f(lo)m(w)i(degree,)f(suc)m(h)f (as)h(the)f(7-cycle.)291 4232 y(Finally)-8 b(,)39 b(let)f(us)d(note)i (that)g(the)f(graphs)g(used)f(in)h(the)h(pro)s(of)e(of)h(Theorems)f(18) i(and)f(19)h(are)g(not)150 4345 y(connected.)j(They)25 b(w)m(ere)g(later)i(made)d(connected)i(via)g(sp)s(ecial)g(gadgets)h(in) d(Section)j(4.5.)40 b(This)24 b(suggest)150 4458 y(that)30 b(p)s(erhaps)e(prior)h(to)h(applying)g(an)f(SDP)-8 b(,)30 b(a)g(graph)f(should)f(b)s(e)h(decomp)s(osed)g(in)m(to)h(comp)s(onen)m (ts)f(of)150 4571 y(strong)g(connectivit)m(y)-8 b(,)33 b(or)c(more)f(generally)-8 b(,)31 b(cut)f(along)g(sparse)f(cuts.)40 b(It)29 b(w)m(ould)g(b)s(e)g(in)m(teresting)h(to)g(see)150 4684 y(whether)g(the)g(idea)h(of)g(\014rst)e(\014nding)g(sparse)h(cuts) h(is)f(helpful)g(in)g(\014nding)f(dense)h(cuts.)1905 5247 y(26)p eop %%Page: 27 27 27 26 bop 150 -58 a Fb(Ac)m(kno)m(wledgemen)m(ts)150 114 y Fn(The)42 b(\014rst)f(author)h(is)g(the)h(Incum)m(b)s(en)m(t)d (of)j(the)f(Joseph)g(and)f(Celia)i(Reskin)f(Career)h(Dev)m(elopmen)m(t) 150 226 y(Chair.)76 b(P)m(art)43 b(of)g(his)e(w)m(ork)i(is)f(supp)s (orted)f(b)m(y)h(a)h(Minerv)-5 b(a)42 b(gran)m(t,)47 b(pro)5 b(ject)43 b(n)m(um)m(b)s(er)d(8354)k(at)f(the)150 339 y(W)-8 b(eizmann)27 b(Institute.)40 b(The)27 b(second)g(author)g (is)g(the)h(William)f(P)m(etsc)m(hek)i(Professor)e(of)h(Mathematics.) 150 452 y(His)j(researc)m(h)g(is)f(partially)h(supp)s(orted)e(b)m(y)h (a)h(gran)m(t)g(of)g(the)f(Israel)h(Science)g(F)-8 b(oundation.)291 565 y(W)g(e)24 b(thank)f(Ho)m(w)m(ard)g(Karlo\013,)i(Mic)m(hael)g (Langb)s(erg,)f(Guen)m(ter)g(Rote)f(and)g(the)g(referees)g(for)f Ff(R)-5 b(andom)150 678 y(Structur)g(es)33 b(and)h(A)n(lgorithms)d Fn(for)g(their)f(commen)m(ts)f(on)i(earlier)g(v)m(ersions)g(of)f(the)h (pap)s(er.)150 964 y Fo(References)195 1167 y Fn([1])47 b(N.)30 b(Alon)g(and)f(B.)h(Sudak)m(o)m(v.)g(\\Bipartite)i(subgraphs)c (and)h(the)h(smallest)f(eigen)m(v)-5 b(alue.)32 b Ff(Combina-)337 1280 y(torics,)h(Pr)-5 b(ob)g(ability)34 b(&)e(Computing)p Fn(,)g(9:1{12,)h(2000.)195 1468 y([2])47 b(F.)41 b(Barahona)g(and)f(A.) h(Mahjoub.)f(\\On)g(the)g(cut)h(p)s(olytop)s(e".)g Ff(Mathematic)-5 b(al)44 b(Pr)-5 b(o)g(gr)g(amming)p Fn(,)337 1581 y(36:157{173,)35 b(1986.)195 1769 y([3])47 b(A.)36 b(Baernstein,)i(B.)e(A.)g(T)-8 b(a)m(ylor.)37 b(\\Spherical)f(rearrangemen)m(ts,)h(subharmonic)c (functions,)k(and)337 1881 y Fl(\003)p Fn(-functions)31 b(in)f Fm(n)p Fn(-space".)h Ff(Duke)h(Math.)h(J.)d Fn(43:245{268,)35 b(1976.)195 2069 y([4])47 b(Y.)30 b(Ben)m(y)m(amini.)h(\\Tw)m(o)g(p)s (oin)m(t)f(symmetrization,)g(the)g(isop)s(erimetric)g(inequalit)m(y)h (on)f(the)h(sphere)337 2182 y(and)25 b(some)g(applications".)i Ff(L)-5 b(onghorn)30 b(Notes,)g(Univ.)d(of)h(T)-7 b(exas,)30 b(T)-7 b(exas)29 b(F)-7 b(unct.)28 b(A)n(nal.)g(Seminar,)337 2295 y Fn(53{76,)33 b(1983-1984.)195 2482 y([5])47 b(M.)31 b(Deza)h(and)e(M.)g(Lauren)m(t.)h Ff(Ge)-5 b(ometry)34 b(of)f(Cuts)g(and)h(Metrics.)c Fn(Springer,)f(1997.)195 2670 y([6])47 b(U.)41 b(F)-8 b(eige)43 b(and)d(M.)i(Go)s(emans.)e (\\Appro)m(ximating)h(the)g(v)-5 b(alue)41 b(of)g(t)m(w)m(o)h(pro)m(v)m (er)g(pro)s(of)e(systems,)337 2783 y(with)d(applications)i(to)g(MAX)f (2-SA)-8 b(T)38 b(and)f(MAX)i(DICUT".)f(In)f Ff(Pr)-5 b(o)g(c)g(e)g(e)g(dings)41 b(of)f(thir)-5 b(d)41 b(Isr)-5 b(ael)337 2896 y(Symp)g(osium)34 b(on)g(The)-5 b(ory)34 b(of)f(Computing)g(and)h(Systems)p Fn(,)d(1995,)i(182{189.)195 3084 y([7])47 b(U.)40 b(F)-8 b(eige,)45 b(M.)c(Karpinski)e(and)g(M.)h (Langb)s(erg.)g(\\Impro)m(v)m(ed)g(Appro)m(ximation)f(of)i(MAX-CUT)337 3196 y(on)32 b(Graphs)f(of)h(Bounded)g(Degree".)i Ff(Ele)-5 b(ctr)g(onic)35 b(Col)5 b(lo)-5 b(quium)35 b(on)g(Computational)i (Complexity)337 3309 y Fn(ECCC)29 b(021,)j(2000.)195 3497 y([8])47 b(M.)34 b(Go)s(emans)g(and)f(D.)i(Williamson.)f(\\Impro)m (v)m(ed)g(appro)m(ximation)f(algorithms)h(for)g(maxim)m(um)337 3610 y(cut)28 b(and)g(satis\014abilit)m(y)i(problems)c(using)i (semide\014nite)g(programming".)e Ff(Journal)32 b(of)f(the)g(A)n(CM)p Fn(,)337 3723 y(42,)g(1115-1145,)k(1995.)195 3910 y([9])47 b(O.)28 b(Goldreic)m(h,)j(S.)d(Goldw)m(asser,)i(D.)g(Ron.)e(\\Prop)s (ert)m(y)h(testing)h(and)e(its)h(connection)h(to)g(learning)337 4023 y(and)g(appro)m(ximation".)h Ff(JA)n(CM)e Fn(45\(4\),)j(653{750,)i (1998.)150 4211 y([10])47 b(J.)22 b(Hastad.)i(\\Some)e(optimal)g (inappro)m(ximabilit)m(y)h(results".)g(In)e Ff(Pr)-5 b(o)g(c)g(e)g(e)g(dings)28 b(of)d(the)h(29th)h(A)n(nnual)337 4324 y(A)n(CM)k(Symp)-5 b(osium)35 b(on)e(The)-5 b(ory)34 b(of)f(Computing)p Fn(,)f(1997,)g(1-10.)150 4512 y([11])47 b(W.)21 b(Ho)s(e\013ding.)g(\\Probabilit)m(y)h(inequalities)g(for)e (sums)f(of)i(b)s(ounded)d(random)h(v)-5 b(ariables".)22 b Ff(Journal)337 4624 y(of)33 b(the)g(A)n(meric)-5 b(an)33 b(Statistic)-5 b(al)34 b(Asso)-5 b(ciation)p Fn(,)32 b(58:13{30,)i(1963.)150 4812 y([12])47 b(H.)31 b(Karlo\013.)h(\\Ho)m(w) h(go)s(o)s(d)e(is)g(the)g(Go)s(emans-Williamson)h(MAX)f(CUT)g (algorithm?".)h Ff(SIAM)g(J.)337 4925 y(Comput.)p Fn(,)g(V)-8 b(ol.)31 b(29,)h(No.)f(1,)g(pp.336{350,)i(1999.)1905 5247 y(27)p eop %%Page: 28 28 28 27 bop 150 -58 a Fn([13])47 b(S.)25 b(Maha)5 b(jan)27 b(and)e(H.)h(Ramesh.)f(\\Derandomizing)h(semide\014nite)f(programming)e (based)j(appro)m(x-)337 55 y(imation)32 b(algorithms".)h(In)e Ff(Pr)-5 b(o)g(c)g(e)g(e)g(dings)37 b(of)d(the)h(36th)h(A)n(nnual)e (IEEE)f(Symp)-5 b(osium)36 b(of)f(F)-7 b(ounda-)337 168 y(tions)33 b(of)g(Computer)h(Scienc)-5 b(e)p Fn(,)30 b(162{169,)k(1995.)150 355 y([14])47 b(P)-8 b(.)34 b(Mattila.)i Ff(Ge)-5 b(ometry)37 b(of)e(sets)h(and)g(me)-5 b(asur)g(es)37 b(in)f(Euclide)-5 b(an)35 b(sp)-5 b(ac)g(es.)37 b(F)-7 b(r)i(actals)38 b(and)e(r)-5 b(e)g(cti\014a-)337 468 y(bility.)31 b Fn(Cam)m(bridge)e(Studies)h(in)h(Adv)-5 b(anced)30 b(Mathematics,)j(44.)e(Cam)m(bridge)f(Univ)m(ersit)m(y)i (Press,)337 581 y(Cam)m(bridge,)d(1995)150 769 y([15])47 b(C.)24 b(McDiarmid.)g(\\On)g(the)g(metho)s(d)e(of)i(b)s(ounded)e (di\013erences".)j(In)f(J.)g(Siemons,)g(editor,)i Ff(Surveys)337 882 y(in)34 b(Combinatorics:)46 b(Invite)-5 b(d)34 b(Pap)-5 b(ers)36 b(at)e(the)g(12th)i(British)e(Combinatorial)j(Confer)-5 b(enc)g(e)p Fn(,)33 b(148{)337 995 y(188.)f(Cam)m(bridge)d(Univ)m (ersit)m(y)j(Press,)e(1989.)150 1182 y([16])47 b(V.D.)58 b(Milman,)64 b(G.)58 b(Sc)m(hec)m(h)m(tman.)g(\\Asymptotic)g(theory)f (of)h(\014nite-dimensional)f(normed)337 1295 y(spaces".)27 b(Lecture)g(Notes)g(in)f(Mathematics,)j(1200.)f(Springer-V)-8 b(erlag,)28 b(Berlin-New)f(Y)-8 b(ork,)28 b(1986.)150 1483 y([17])47 b(G.)54 b(Sc)m(hec)m(h)m(tman.)g(\\Concen)m(tration,)62 b(results)53 b(and)g(applications".)j(Handb)s(o)s(ok)c(on)i(Banac)m(h) 337 1596 y(spaces,)31 b(to)g(app)s(ear.)150 1783 y([18])47 b(R.)30 b(Sc)m(hneider.)g(Con)m(v)m(ex)i(Bo)s(dies:)41 b(The)29 b(Brunn-Mink)m(o)m(wski)i(Theory)-8 b(.)30 b(Encyclop)s(edia)h (of)g(Mathe-)337 1896 y(matics)f(and)g(its)h(Applications,)g(V)-8 b(olume)31 b(44,)g(Cam)m(bridge)e(Univ)m(ersit)m(y)j(Press,)e(1993.)150 2084 y([19])47 b(U.)33 b(Zwic)m(k.)g(\\Out)m(w)m(ard)f(rotations:)47 b(a)33 b(to)s(ol)g(for)f(rounding)g(solutions)h(of)f(semide\014nite)g (program-)337 2197 y(ming)25 b(relaxations,)j(with)e(applications)h(to) 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e(the)h(sphere)f Fm(S)2419 3266 y Fk(d)p Fi(\000)p Fj(1)2575 3299 y Fn(of)h(maximal)e(cardinalit)m(y)j(among)150 3412 y(all)d(sets)g(with)f(the)h(prop)s(ert)m(y)e(that)i(the)g(distance)g(b) s(et)m(w)m(een)g(an)m(y)f(t)m(w)m(o)i(v)m(ectors)g(is)e(at)h(least)h Fm(\015)5 b(=)p Fn(4.)39 b(Consider)150 3525 y(the)c(graph)f(whose)g(v) m(ertices)i(are)f(the)f(n)m(um)m(b)s(ers)e(1)p Fm(;)15 b(:)g(:)g(:)j(;)d(m)34 b Fn(and)g(\()p Fm(i;)15 b(j)5 b Fn(\))36 b(is)f(an)f(edge)h(if)g(and)f(only)g(if)h(the)150 3638 y(distance)h(b)s(et)m(w)m(een)g Fm(u)917 3652 y Fk(i)980 3638 y Fn(and)e Fm(u)1213 3652 y Fk(j)1285 3638 y Fn(is)h(exactly)i Fm(\015)5 b(=)p Fn(4.)56 b(By)36 b(mo)m(ving)e(some)h(of)g(the)h Fm(u)2947 3652 y Fk(i)2975 3638 y Fn(-s)f(w)m(e)h(can)f(and)g(shall)150 3751 y(assume)j(that)h (the)g(graph)f(is)h(connected)h(\(this)f(is)g(done)g(only)g(to)g(sligh) m(tly)i(impro)m(v)m(e)d(the)h(n)m(umerical)150 3864 y(constan)m(ts)27 b(b)s(elo)m(w\).)40 b(Let)26 b Fm(T)39 b Fn(b)s(e)25 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Fn(of)h(v)m(olume)f(at)i(most)e(1)p Fm(=)-5 b(N)45 b Fn(suc)m(h)34 b(that)i(the)f(v)m(olume)f(of)h Fm(C)2867 5007 y Fk(i)2927 4993 y Fn(=)d Fm(V)3083 5007 y Fk(i)3135 4993 y Fl(n)23 b Fm(U)3265 5007 y Fk(i)3328 4993 y Fn(is)35 b(divisible)1905 5247 y(28)p eop %%Page: 29 29 29 28 bop 150 -58 a Fn(b)m(y)28 b(1)p Fm(=)-5 b(N)10 b Fn(.)41 b(Lo)s(ok)28 b(no)m(w)h(at)g(the)f(next)h(lev)m(el)h(of)e (the)h(tree)g(\(the)f(v)m(ertices)i(whic)m(h)e(are)h(connected)g(to)g (lea)m(v)m(es\))150 55 y(and)e(for)h(eac)m(h)h(of)f(them)f(de\014ne)g (ten)m(tativ)m(ely)k Fm(C)1778 22 y Fi(0)1771 79 y Fk(i)1826 55 y Fn(=)25 b Fm(V)1975 69 y Fk(i)2018 55 y Fl([)15 b([)p Fm(U)2217 69 y Fk(j)2281 55 y Fn(where)27 b(the)h(last)h(union)e (is)h(o)m(v)m(er)h(all)g(lea)m(v)m(es)150 168 y Fm(j)36 b Fn(suc)m(h)30 b(that)h(\()p Fm(j;)15 b(i)p Fn(\))33 b(is)d(an)h(edge.)42 b(let)31 b Fm(U)1490 182 y Fk(i)1544 168 y Fl(\032)25 b Fm(V)1693 182 y Fk(i)1752 168 y Fn(b)s(e)30 b(of)g(measure)g(at)h(most)f(1)p Fm(=)-5 b(N)41 b 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b(\\su\016cien)m (t")i(n)m(umerical)150 1410 y(precision)31 b(the)g(cells)g(can)g(b)s(e) f(made)g(equal)h(up)e(to)j(an)e(error)g(that)i(is)e(negligible)j (compared)c(to)i Fm(\017)p Fn(.)42 b(This)150 1523 y(will)31 b(su\016ce)f(for)g(pro)s(ducing)f(a)i(graph)f Fm(G)f Fn(with)i(in)m(tegralit)m(y)i(ratio)e(b)s(elo)m(w)g Fm(\013)20 b Fn(+)g Fm(\017)p Fn(.)291 1681 y Fg(Pro)s(of)36 b(of)f(Theorem)g(5:) 72 b Fn(Giv)m(en)31 b(a)g(h)m(yp)s(erplane)f Fm(H)37 b Fn(through)30 b(the)h(origin)f(w)m(e)h(denote)h(b)m(y)e Fm(S)3584 1695 y Fj(0)3654 1681 y Fn(its)150 1794 y(in)m(tersection)f (with)e(the)h(sphere)f Fm(S)1335 1761 y Fk(d)p Fi(\000)p Fj(1)1465 1794 y Fn(,)i(b)m(y)e Fm(S)1698 1808 y Fj(+)1784 1794 y Fn(one)h(of)g(the)f(t)m(w)m(o)i(half)f(spheres)e(remaining)h(in) g Fm(S)3450 1761 y Fk(d)p Fi(\000)p Fj(1)3595 1794 y Fl(n)14 b Fm(S)3710 1808 y Fj(0)150 1907 y Fn(and)36 b(b)m(y)h Fm(S)522 1921 y Fi(\000)618 1907 y Fn(the)h(other)f(half)g (sphere.)60 b(W)-8 b(e)38 b(also)g(denote)g(b)m(y)f Fm(\033)j Fn(=)c Fm(\033)2587 1921 y Fk(H)2691 1907 y Fn(the)h(re\015ection)h (with)f(resp)s(ect)150 2020 y(to)g Fm(H)7 b Fn(.)57 b(\(F)-8 b(or)37 b(example,)g(if)f Fm(H)43 b Fn(consists)36 b(of)g(all)h(v)m (ectors)g(whose)f(\014rst)f(comp)s(onen)m(t)g(is)h(zero,)i(then)e Fm(\033)j Fn(is)150 2132 y(the)32 b(map)f(whic)m(h)h(c)m(hanges)h(the)g (sign)f(of)g(the)g(\014rst)g(comp)s(onen)m(t)f(of)i(eac)m(h)g(v)m (ector)h(not)e(in)g Fm(H)39 b Fn(and)32 b(is)g(the)150 2245 y(iden)m(tit)m(y)37 b(on)e Fm(H)7 b Fn(.\))57 b(Note)37 b(that)f(for)f(an)m(y)h(\(measurable\))g(set)g Fm(A)e Fl(\032)f Fm(S)2536 2212 y Fk(d)p Fi(\000)p Fj(1)2667 2245 y Fn(,)k Fm(\026)p Fn(\()p Fm(A)p Fn(\))d(=)g Fm(\026)p Fn(\()p Fm(\033)s Fn(\()p Fm(A)p Fn(\)\))j(and)e(also)150 2358 y(that)i(if)f Fm(x)g Fn(is)g(in)f Fm(S)794 2372 y Fi(\000)889 2358 y Fn(and)h Fm(y)i Fn(in)e Fm(S)1323 2372 y Fj(+)1418 2358 y Fn(then)g(the)g(distance)h(b)s(et)m(w)m(een)f Fm(x)g Fn(and)g Fm(\033)s Fn(\()p Fm(y)s Fn(\))g(is)g(smaller)g(than)g (the)150 2471 y(distance)31 b(b)s(et)m(w)m(een)g Fm(x)f Fn(and)g Fm(y)s Fn(.)291 2584 y(Giv)m(en)40 b(a)f(subset)f Fm(A)h Fn(of)h Fm(S)1215 2551 y Fk(d)p Fi(\000)p Fj(1)1384 2584 y Fn(w)m(e)f(de\014ne)f(its)i(symmetrization)e Fm(A)2654 2551 y Fi(\003)2732 2584 y Fn(with)h(resp)s(ect)g(to)g Fm(H)46 b Fn(in)39 b(the)150 2697 y(follo)m(wing)32 b(w)m(a)m(y:)41 b(If)30 b Fm(x)g Fn(is)g(a)g(p)s(oin)m(t)g(in)g Fm(A)20 b Fl(\\)f Fm(S)1659 2711 y Fi(\000)1748 2697 y Fn(suc)m(h)29 b(that)i Fm(\033)s Fn(\()p Fm(x)p Fn(\))g(is)f(not)g(in)g Fm(A)g Fn(then)g(w)m(e)h(include)e Fm(\033)s Fn(\()p Fm(x)p Fn(\))i(in)150 2810 y Fm(A)218 2777 y Fi(\003)286 2810 y Fn(and)c(do)h(not)h(include)f Fm(x)g Fn(in)g Fm(A)1307 2777 y Fi(\003)1346 2810 y Fn(.)40 b(All)29 b(other)f(p)s(oin)m(ts)h (of)f Fm(A)g Fn(will)h(b)s(e)e(also)i(in)f Fm(A)2905 2777 y Fi(\003)2973 2810 y Fn(and)g(no)g(other)g(p)s(oin)m(ts)150 2923 y(will)j(b)s(e)e(added)h(to)h Fm(A)892 2890 y Fi(\003)932 2923 y Fn(.)41 b(F)-8 b(ormally)g(,)520 3114 y Fm(A)588 3076 y Fi(\003)653 3114 y Fn(=)25 b([)p Fm(A)c Fl(\\)f Fn(\()p Fm(S)1035 3128 y Fj(+)1114 3114 y Fl([)g Fm(S)1251 3128 y Fj(0)1290 3114 y Fn(\)])h Fl([)f Fn([)p Fm(A)g Fl(\\)g Fm(S)1702 3128 y Fi(\000)1781 3114 y Fl(\\)g Fm(\033)s Fn(\()p Fm(A)h Fl(\\)f Fm(S)2178 3128 y Fj(+)2236 3114 y Fn(\)])h Fl([)f Fn([)p Fm(\033)s Fn(\()p Fm(A)h Fl(\\)f Fm(S)2739 3128 y Fi(\000)2818 3114 y Fl(n)h Fm(\033)s Fn(\()p Fm(A)g Fl(\\)f Fm(S)3200 3128 y Fj(+)3258 3114 y Fn(\)\)])p Fm(:)150 3304 y Fn(The)25 b(p)s(oin)m(t)g(is)g(that)h(b)m (y)f(\\pushing)g(p)s(oin)m(ts)g(of)g Fm(A)g Fn(up")g(whenev)m(er)g(p)s (ossible,)h Fm(A)g Fn(b)s(ecomes)e(closer)i(to)g(b)s(eing)150 3417 y(a)31 b(cap.)41 b(Note)31 b(that)g Fm(\026)p Fn(\()p Fm(A)1002 3384 y Fi(\003)1042 3417 y Fn(\))f(is)h(equal)f(to)h Fm(\026)p Fn(\()p Fm(A)p Fn(\).)42 b(W)-8 b(e)31 b(shall)g(next)f(sho)m (w)g(that)h Fm(\026)2859 3431 y Fk(\032)2899 3417 y Fn(\()p Fm(A)3002 3384 y Fi(\003)3042 3417 y Fn(\))g(is)f(larger)h(than)f(or) 150 3530 y(equal)h(to)g Fm(\026)556 3544 y Fk(\032)596 3530 y Fn(\()p Fm(A)p Fn(\).)291 3643 y(W)-8 b(e)40 b(presen)m(t)f(the) g(basic)h(idea)f(of)g(the)g(pro)s(of)g(that)g Fm(\026)2167 3657 y Fk(\032)2207 3643 y Fn(\()p Fm(A)2310 3610 y Fi(\003)2350 3643 y Fn(\))h Fl(\025)f Fm(\026)2590 3657 y Fk(\032)2630 3643 y Fn(\()p Fm(A)p Fn(\).)67 b(W)-8 b(e)41 b(sa)m(y)e(that)h(a)f (pair)g(of)150 3756 y(p)s(oin)m(ts)27 b Fm(x;)15 b(y)30 b Fn(con)m(tributes)e(to)f Fm(\026)1219 3770 y Fk(\032)1259 3756 y Fn(\()p Fm(A)p Fn(\))h(\(resp)s(ectiv)m(ely)-8 b(,)30 b(to)e Fm(\026)2134 3770 y Fk(\032)2174 3756 y Fn(\()p Fm(A)2277 3723 y Fi(\003)2317 3756 y Fn(\)\))g(if)f(exactly)i (one)e(of)g(the)g(p)s(oin)m(ts)g(is)g(in)g Fm(A)150 3869 y Fn(\(resp)s(ectiv)m(ely)-8 b(,)29 b Fm(A)763 3836 y Fi(\003)802 3869 y Fn(\))d(and)e Fm(d)p Fn(\()p Fm(x;)15 b(y)s Fn(\))27 b Fl(\025)e Fm(\032)p Fn(.)38 b(W)-8 b(e)27 b(shall)e(sho)m(w)g(a)h(one-to-one)h(mapping)c(from)h(pairs)h(of)g(p)s (oin)m(ts)150 3982 y(con)m(tributing)h(to)h Fm(\026)825 3996 y Fk(\032)865 3982 y Fn(\()p Fm(A)p Fn(\))g(to)g(pairs)e(of)h(p)s (oin)m(ts)g(con)m(tributing)g(to)h Fm(\026)2398 3996 y Fk(\032)2438 3982 y Fn(\()p Fm(A)2541 3949 y Fi(\003)2581 3982 y Fn(\).)40 b(In)m(tuitiv)m(ely)-8 b(,)29 b(this)d(means)e(that) 150 4095 y Fm(\026)205 4109 y Fk(\032)245 4095 y Fn(\()p Fm(A)348 4062 y Fi(\003)388 4095 y Fn(\))32 b(is)g(larger.)47 b(T)-8 b(ec)m(hnically)34 b(this)d(is)h(not)h(a)f(rigorous)g(argumen)m (t,)g(b)s(ecause)g(the)g(n)m(um)m(b)s(er)e(of)i(pairs)150 4208 y(of)i(p)s(oin)m(ts)f(in)h(in\014nite)f(\(and)h(uncoun)m(table\).) 51 b(Ho)m(w)m(ev)m(er,)37 b(in)c(our)g(case)i(it)f(can)g(b)s(e)f(made)g (rigorous)h(\(b)m(y)150 4320 y(partitioning)40 b(pairs)f(of)h(p)s(oin)m (ts)f(in)m(to)h(a)g(\014nite)f(n)m(um)m(b)s(er)e(of)i(sets)h(and)f (arguing)g(with)g(resp)s(ect)h(to)g(the)150 4433 y(measure)29 b(of)i(eac)m(h)g(set)g(separately\),)h(though)f(details)g(are)g (omitted.)291 4546 y(Consider)26 b(\014rst)g(a)h(p)s(oin)m(t)f Fm(x)f Fl(2)g Fm(S)1375 4560 y Fj(0)1414 4546 y Fn(.)40 b(W)-8 b(e)27 b(claim)g(that)g(the)g(con)m(tribution)g(of)g(pairs)f(of) h(p)s(oin)m(ts)f(in)g(whic)m(h)150 4659 y(it)31 b(participates)h(to)f Fm(\026)903 4673 y Fk(\032)943 4659 y Fn(\()p Fm(A)1046 4626 y Fi(\003)1086 4659 y Fn(\))g(is)f(iden)m(tical)i(to)f(the)g(con)m (tribution)g(to)g Fm(\026)2561 4673 y Fk(\032)2601 4659 y Fn(\()p Fm(A)p Fn(\).)42 b(Note)31 b(\014rst)f(that)h Fm(x)25 b Fl(2)g Fm(A)31 b Fn(i\013)150 4772 y Fm(x)25 b Fl(2)g Fm(A)381 4739 y Fi(\003)421 4772 y Fn(.)40 b(As)30 b(this)f(holds)g(for)g(an)m(y)h(other)g(p)s(oin)m(t)g Fm(y)e Fl(2)d Fm(S)2031 4786 y Fj(0)2070 4772 y Fn(,)30 b(pairs)f Fm(x;)15 b(y)33 b Fn(b)s(oth)c(in)g Fm(S)2892 4786 y Fj(0)2960 4772 y Fn(con)m(tribute)i(the)f(same)150 4885 y(to)e Fm(\026)313 4899 y Fk(\032)353 4885 y Fn(\()p Fm(A)456 4852 y Fi(\003)496 4885 y Fn(\))f(and)g Fm(\026)787 4899 y Fk(\032)827 4885 y Fn(\()p Fm(A)p Fn(\).)40 b(Consider)26 b(no)m(w)h(a)h(p)s(oin)m(t)f Fm(y)h Fl(62)d Fm(S)2114 4899 y Fj(0)2153 4885 y Fn(.)39 b(The)27 b(idea)g(is)h(to)f(consider)g (also)h Fm(\033)s Fn(\()p Fm(y)s Fn(\))g(at)g(the)150 4998 y(same)34 b(time.)52 b(No)m(w)36 b Fm(d)p Fn(\()p Fm(x;)15 b(y)s Fn(\))33 b(=)e Fm(d)p Fn(\()p Fm(x;)15 b(\033)s Fn(\()p Fm(y)s Fn(\)\),)38 b(and)c Fl(jf)p Fm(y)s(;)15 b(\033)s Fn(\()p Fm(y)s Fn(\))p Fl(g)2257 4934 y Fe(T)2344 4998 y Fm(A)p Fl(j)32 b Fn(=)g Fl(jf)p Fm(y)s(;)15 b(\033)s Fn(\()p Fm(y)s Fn(\))p Fl(g)2963 4934 y Fe(T)3049 4998 y Fm(A)3117 4965 y Fi(\003)3157 4998 y Fl(j)p Fn(.)53 b(This)34 b(implies)1905 5247 y(29)p eop %%Page: 30 30 30 29 bop 150 -58 a Fn(that)35 b(the)f(t)m(w)m(o)h(pairs)f Fm(x;)15 b(y)37 b Fn(and)d Fm(x;)15 b(\033)s Fn(\()p Fm(y)s Fn(\))35 b(con)m(tribute)g(the)f(same)f(to)i Fm(\026)2570 -44 y Fk(\032)2610 -58 y Fn(\()p Fm(A)2713 -91 y Fi(\003)2753 -58 y Fn(\))f(and)g Fm(\026)3058 -44 y Fk(\032)3098 -58 y Fn(\()p Fm(A)p Fn(\).)53 b(Going)35 b(o)m(v)m(er)150 55 y(all)c Fm(y)i Fn(w)m(e)e(ha)m(v)m(e)h(the)e(same)g(claim)h(for)f Fm(x)p Fn(.)291 168 y(Consider)h(no)m(w)i(a)f(p)s(oin)m(t)g Fm(x)1232 182 y Fj(+)1319 168 y Fl(2)c Fm(S)1464 182 y Fj(+)1555 168 y Fn(and)j Fm(x)1785 182 y Fi(\000)1873 168 y Fn(=)c Fm(\033)s Fn(\()p Fm(x)2113 182 y Fj(+)2173 168 y Fn(\).)46 b(As)32 b Fl(jf)p Fm(x)2537 182 y Fj(+)2597 168 y Fm(;)15 b(x)2689 182 y Fi(\000)2748 168 y Fl(g)2808 103 y Fe(T)2893 168 y Fm(A)p Fl(j)29 b Fn(=)f Fl(jf)p Fm(x)3236 182 y Fj(+)3296 168 y Fm(;)15 b(x)3388 182 y Fi(\000)3447 168 y Fl(g)3507 103 y Fe(T)3592 168 y Fm(A)3660 135 y Fi(\003)3699 168 y Fl(j)p Fn(,)150 281 y(the)31 b(pair)f Fm(x)546 295 y Fj(+)605 281 y Fm(;)15 b(x)697 295 y Fi(\000)786 281 y Fn(con)m(tributes)31 b(the)g(same)f(to)h Fm(\026)1810 295 y Fk(\032)1850 281 y Fn(\()p Fm(A)1953 248 y Fi(\003)1993 281 y Fn(\))g(and)e Fm(\026)2290 295 y Fk(\032)2330 281 y Fn(\()p Fm(A)p Fn(\).)291 394 y(It)34 b(remains)e(only)i(to)h(consider)e(pairs)g Fm(x;)15 b(y)37 b Fn(where)c(neither)h(p)s(oin)m(t)g(is)g(in)f Fm(S)2889 408 y Fj(0)2962 394 y Fn(and)g Fm(x)d Fl(6)p Fn(=)h Fm(\033)s Fn(\()p Fm(y)s Fn(\).)51 b(Also)150 506 y(here)31 b(the)h(idea)g(is)g(to)g(consider)f(them)g(together)i (with)e(their)h(re\015ections.)45 b(So)31 b(consider)g Fm(x)3283 520 y Fj(+)3370 506 y Fl(2)26 b Fm(S)3513 520 y Fj(+)3604 506 y Fn(and)150 619 y Fm(x)202 633 y Fi(\000)297 619 y Fn(=)37 b Fm(\033)s Fn(\()p Fm(x)547 633 y Fj(+)606 619 y Fn(\),)j(and)c Fm(y)934 633 y Fj(+)1029 619 y Fl(2)g Fm(S)1182 633 y Fj(+)1278 619 y Fn(and)h Fm(y)1507 633 y Fi(\000)1602 619 y Fn(=)f Fm(\033)s Fn(\()p Fm(y)1844 633 y Fj(+)1903 619 y Fn(\).)61 b(They)37 b(con)m(tribute)h(4)f (previously)g(unconsidered)150 732 y(pairs,)e(namely)e Fm(x)774 746 y Fj(+)833 732 y Fm(;)15 b(y)918 746 y Fj(+)1011 732 y Fn(and)33 b Fm(x)1243 746 y Fj(+)1302 732 y Fm(;)15 b(y)1387 746 y Fi(\000)1480 732 y Fn(and)34 b Fm(x)1713 746 y Fi(\000)1772 732 y Fm(;)15 b(y)1857 746 y Fj(+)1949 732 y Fn(and)34 b Fm(x)2182 746 y Fi(\000)2241 732 y Fm(;)15 b(y)2326 746 y Fi(\000)2385 732 y Fn(.)51 b(Of)34 b(these)g(four)f(pairs,)i(it)g(is)f(easy)g(to)150 845 y(see)c(that)f(the)h(n)m(um)m(b)s(er)d(of)i(pairs)g(cut)g(b)m(y)g Fm(A)g Fn(is)h(equal)f(to)h(the)f(n)m(um)m(b)s(er)e(of)j(pairs)e(cut)i (b)m(y)f Fm(A)3275 812 y Fi(\003)3314 845 y Fn(.)41 b(Note)30 b(also)150 958 y(that)j Fm(d)p Fn(\()p Fm(x)483 972 y Fj(+)542 958 y Fm(;)15 b(y)627 972 y Fj(+)686 958 y Fn(\))29 b(=)f Fm(d)p Fn(\()p Fm(x)983 972 y Fi(\000)1042 958 y Fm(;)15 b(y)1127 972 y Fi(\000)1186 958 y Fn(\))29 b Fm(<)f(d)p Fn(\()p Fm(x)1483 972 y Fj(+)1542 958 y Fm(;)15 b(y)1627 972 y Fi(\000)1686 958 y Fn(\))29 b(=)e Fm(d)p Fn(\()p Fm(x)1982 972 y Fi(\000)2042 958 y Fm(;)15 b(y)2127 972 y Fj(+)2186 958 y Fn(\).)46 b(A)32 b(simple)g(case)h (analysis)f(sho)m(ws)g(that)h(of)150 1071 y(the)k(four)g(pairs,)h Fm(A)834 1038 y Fi(\003)911 1071 y Fn(alw)m(a)m(ys)g(cuts)f(pairs)g (that)h(are)f(at)h(least)g(as)f(far)g(apart)g(as)h(the)f(pairs)f(cut)i (b)m(y)f Fm(A)p Fn(.)150 1184 y(Hence)c(the)g(con)m(tribution)g(to)g Fm(\026)1269 1198 y Fk(\032)1309 1184 y Fn(\()p Fm(A)1412 1151 y Fi(\003)1452 1184 y Fn(\))g(cannot)f(b)s(e)g(smaller)g(than)g (that)h(to)g Fm(\026)2842 1198 y Fk(\032)2882 1184 y Fn(\()p Fm(A)p Fn(\).)48 b(This)32 b(\014nishes)f(the)150 1297 y(\(sk)m(etc)m(h)h(of)7 b(\))31 b(pro)s(of)f(that)h Fm(\026)1097 1311 y Fk(\032)1137 1297 y Fn(\()p Fm(A)1240 1264 y Fi(\003)1280 1297 y Fn(\))25 b Fl(\025)g Fm(\026)1491 1311 y Fk(\032)1531 1297 y Fn(\()p Fm(A)p Fn(\).)291 1410 y(The)i(rest)g(of)h(the)f(pro)s(of)f(is)i(v)m(ery)f(similar)g(to)h (the)f(one)h(exp)s(osed)f(in)g([17)q(].)40 b(The)26 b(idea)i(is)f(to)h (sho)m(w)f(that)150 1523 y(a)34 b(rep)s(etitiv)m(e)h(op)s(erations)f (of)g(the)g(kind)f(ab)s(o)m(v)m(e)i(with)e(man)m(y)g(di\013eren)m(t)h (h)m(yp)s(erplanes)f(reduces)g(the)h(set)150 1636 y Fm(A)c Fn(to)i(a)e(cap.)41 b(Some)30 b(classical)i(Mathematics)g(needs)e(to)h (b)s(e)f(used)f(here.)291 1748 y(Consider)38 b(the)h(metric)g(space)g Fl(C)44 b Fn(of)39 b(all)h(closed)f(subsets)g(of)g Fm(S)2499 1715 y Fk(d)p Fi(\000)p Fj(1)2668 1748 y Fn(with)g(the)g(Hausdor\013)f (metric.)150 1861 y(\(Tw)m(o)28 b(sets)g Fm(A)f Fn(and)g Fm(B)32 b Fn(are)c(of)f(distance)h(at)h(most)d Fm(\017)i Fn(if)f(for)g(all)h Fm(a)e Fl(2)f Fm(A)i Fn(there)h(is)f(a)h Fm(b)d Fl(2)g Fm(B)32 b Fn(whose)27 b(distance)150 1974 y(from)37 b Fm(a)h Fn(is)g(at)h(most)f Fm(\017)g Fn(and)f(the)i(same)e (holds)h(when)f(in)m(terc)m(hanging)j(the)e(roles)h(of)f Fm(A)h Fn(and)e Fm(B)5 b Fn(.\))64 b(Fix)150 2087 y Fm(A)25 b Fl(2)g(C)36 b Fn(and)30 b(consider)g(the)g(set)h Fl(B)d(\022)d(C)35 b Fn(of)c(all)g(sets)g Fm(B)f Fl(2)25 b(C)35 b Fn(satisfying:)286 2298 y Fl(\017)46 b Fm(\026)p Fn(\()p Fm(B)5 b Fn(\))26 b(=)e Fm(\026)p Fn(\()p Fm(A)p Fn(\))286 2485 y Fl(\017)46 b Fn(F)-8 b(or)31 b(all)h(0)25 b Fm(<)g(\032)h(<)f(\031)94 b(\026)1203 2499 y Fk(\032)1243 2485 y Fn(\()p Fm(B)5 b Fn(\))25 b Fl(\025)g Fm(\026)1563 2499 y Fk(\032)1603 2485 y Fn(\()p Fm(A)p Fn(\),)92 b(and)286 2672 y Fl(\017)46 b Fn(for)30 b(all)i Fm(\017)25 b(>)g Fn(0)61 b Fm(\026)p Fn(\()p Fm(B)1066 2686 y Fk(\017)1099 2672 y Fn(\))25 b Fl(\024)g Fm(\026)p Fn(\()p Fm(A)1413 2686 y Fk(\017)1446 2672 y Fn(\).)150 2883 y(Here)31 b Fm(B)434 2897 y Fk(\017)497 2883 y Fn(denotes)f(the)h(set)f(of)h(all)g(p)s(oin)m(ts)f(whose)g (distance)h(to)g(some)e(p)s(oin)m(t)i(in)f Fm(B)k Fn(is)d(at)g(most)e Fm(\017)p Fn(.)41 b(Note)150 2996 y(that)h Fl(B)h Fn(is)e(not)h(empt)m (y)-8 b(,)43 b(as)f(it)f(con)m(tains)i Fm(A)p Fn(.)73 b(Moreo)m(v)m(er,)46 b(the)41 b(symmetrization)f(pro)s(cess)h(preserv)m (es)150 3109 y(mem)m(b)s(ership)f(in)j Fl(B)s Fn(.)80 b(This)43 b(w)m(as)h(already)g(pro)m(v)m(ed)g(for)g(the)g(\014rst)f(t)m (w)m(o)i(prop)s(erties)e(of)g Fl(B)s Fn(.)80 b(F)-8 b(or)45 b(the)150 3222 y(third)36 b(prop)s(ert)m(y)h(w)m(e)g(need)g(to)g(sho)m (w)g(that)h(for)f(ev)m(ery)g(h)m(yp)s(erplane,)h(ev)m(ery)g(set)g Fm(A)f Fn(and)f(ev)m(ery)i Fm(\017)e(>)g Fn(0,)150 3334 y Fm(\026)p Fn(\()p Fm(A)308 3301 y Fi(\003)308 3357 y Fk(\017)348 3334 y Fn(\))42 b Fl(\024)g Fm(\026)p Fn(\()p Fm(A)696 3348 y Fk(\017)729 3334 y Fn(\).)71 b(This)40 b(is)h(kno)m(wn,)h(and)e(used)g(for)g(example)g(in)h(the)f(pro)s(of)g (of)h(the)f(isop)s(erimetric)150 3447 y(inequalit)m(y)30 b(that)f(is)f(review)m(ed)h(in)g([17)q(].)40 b(W)-8 b(e)30 b(added)e(the)g(condition)h Fm(\026)p Fn(\()p Fm(B)2668 3461 y Fk(\017)2701 3447 y Fn(\))d Fl(\024)f Fm(\026)p Fn(\()p Fm(A)3016 3461 y Fk(\017)3049 3447 y Fn(\))j(to)i(the)e (de\014nition)150 3560 y(of)37 b Fl(B)i Fn(in)e(order)f(to)h(ensure)f (that)i Fl(B)h Fn(is)e(closed)g(in)g Fl(C)5 b Fn(;)40 b(i.e.,)g(if)d Fm(B)2368 3574 y Fk(n)2451 3560 y Fn(is)g(a)g(sequence)g (of)g(sets)h(in)e Fl(B)j Fn(whic)m(h)150 3673 y(tends)31 b(in)h(the)f(Hausdor\013)h(metric)f(to)h Fm(B)k Fn(then)c Fm(B)k Fn(is)c(also)g(in)g Fl(B)s Fn(.)44 b(W)-8 b(e)32 b(state)i(it)e(as)g(a)g(lemma)e(and)h(dela)m(y)150 3786 y(the)g(pro)s(of.)150 3997 y Fg(Lemma)k(22)46 b Fl(B)35 b Ff(is)d(close)-5 b(d)34 b(in)f Fl(C)291 4208 y Fn(Fix)26 b(a)f(p)s(oin)m(t)h Fm(x)804 4222 y Fj(0)868 4208 y Fl(2)f Fm(S)1015 4175 y Fk(d)p Fi(\000)p Fj(1)1171 4208 y Fn(and)f(let)i Fm(C)32 b Fn(b)s(e)25 b(the)g(closed)h(cap)f(cen)m(tered)i(at)e Fm(x)2773 4222 y Fj(0)2838 4208 y Fn(with)g(measure)f Fm(\026)p Fn(\()p Fm(A)p Fn(\).)39 b(T)-8 b(o)150 4320 y(conclude)26 b(the)g(pro)s(of)f(it)i(is)f(enough)f(to)i(pro)m(v)m(e)g (that)f Fm(C)32 b Fl(2)25 b(B)s Fn(.)38 b(F)-8 b(or)27 b(an)m(y)f(h)m(yp)s(erplane)f Fm(H)33 b Fn(with)25 b Fm(x)3387 4334 y Fj(0)3462 4320 y Fm(=)-55 b Fl(2)25 b Fm(H)32 b Fn(w)m(e)150 4433 y(denote)26 b(b)m(y)g Fm(S)616 4447 y Fj(+)700 4433 y Fn(the)g(op)s(en)f(half)h(sphere)f(determined)f (b)m(y)h Fm(H)7 b Fn(,)27 b(con)m(taining)g Fm(x)2736 4447 y Fj(0)2776 4433 y Fn(.)39 b Fm(B)30 b Fl(!)25 b Fm(\026)p Fn(\()p Fm(B)15 b Fl(\\)c Fm(C)c Fn(\))25 b(is)h(upp)s(er)150 4546 y(semi)37 b(con)m(tin)m(uous)h(on)g Fl(C)5 b Fn(;)41 b(i.e.,)g(if)d Fm(B)1421 4560 y Fk(n)1505 4546 y Fn(is)g(a)g(sequence)g (of)g(sets)g(in)f Fl(C)43 b Fn(whic)m(h)37 b(tends)g(in)h(the)f (Hausdor\013)150 4659 y(metric)30 b(to)h Fm(B)k Fn(then)30 b Fm(\026)p Fn(\()p Fm(B)25 b Fl(\\)20 b Fm(C)7 b Fn(\))30 b(is)g(at)h(least)h(lim)15 b(sup)e Fm(\026)p Fn(\()p Fm(B)2124 4673 y Fk(n)2192 4659 y Fl(\\)20 b Fm(C)7 b Fn(\).)291 4772 y Fl(C)37 b Fn(with)31 b(the)g(top)s(ology)i(induced)d (b)m(y)i(the)f(Hausdor\013)g(metric)g(is)h(a)f(compact)h(set)g(and)f (since)g Fl(B)j Fn(is)e(a)150 4885 y(closed)h(subset,)f Fm(\026)p Fn(\()p Fm(B)26 b Fl(\\)21 b Fm(C)7 b Fn(\))32 b(ac)m(hiev)m(es)j(its)d(maxim)m(um)d(for)j(some)g Fm(B)h Fl(2)27 b(B)s Fn(.)46 b(It)32 b(is)g(enough)g(to)h(sho)m(w)f(that)150 4998 y Fm(B)40 b Fn(can)35 b(b)s(e)g(tak)m(en)h(to)g(b)s(e)f(equal)g (to)h Fm(C)7 b Fn(.)55 b(If)35 b(this)g(is)g(not)g(the)h(case)g(then)f Fm(\026)p Fn(\()p Fm(B)28 b Fl(n)c Fm(C)7 b Fn(\))33 b(=)g Fm(\026)p Fn(\()p Fm(C)d Fl(n)24 b Fm(B)5 b Fn(\))33 b Fm(>)g Fn(0.)1905 5247 y(30)p eop %%Page: 31 31 31 30 bop 150 -58 a Fn(Denote)27 b(b)m(y)f Fm(B)5 b Fn(\()p Fm(z)t(;)15 b(r)s Fn(\))27 b(the)f(ball)g(of)h(radius)e Fm(r)j Fn(cen)m(tered)f(at)g Fm(z)t Fn(.)39 b(By)27 b(Leb)s(esgue's)f (densit)m(y)g(theorem)f(\(see)i(for)150 55 y(example)j([14)q(],)h(page) g(40\))g(there)g(are)f(p)s(oin)m(ts)h Fm(x)25 b Fl(2)g Fm(B)f Fl(n)d Fm(C)36 b Fn(and)30 b Fm(y)e Fl(2)d Fm(C)h Fl(n)21 b Fm(B)35 b Fn(suc)m(h)30 b(that,)h(for)f(some)f(small)150 168 y Fm(r)f(>)d Fn(0,)j Fm(\026)p Fn(\()p Fm(B)5 b Fn(\()p Fm(x;)15 b(r)s Fn(\))f Fl(\\)g Fn(\()p Fm(B)20 b Fl(n)14 b Fm(C)7 b Fn(\)\))26 b Fm(>)f Fn(0)p Fm(:)p Fn(99)p Fm(\026)p Fn(\()p Fm(B)5 b Fn(\()p Fm(x;)15 b(r)s Fn(\)\))30 b(and)d Fm(\026)p Fn(\()p Fm(B)5 b Fn(\()p Fm(y)s(;)15 b(r)s Fn(\))f Fl(\\)g Fn(\()p Fm(C)21 b Fl(n)14 b Fm(B)5 b Fn(\)\))27 b Fm(>)e Fn(0)p Fm(:)p Fn(99)p Fm(\026)p Fn(\()p Fm(B)5 b Fn(\()p Fm(y)s(;)15 b(r)s Fn(\)\).)41 b(Let)150 281 y Fm(H)i Fn(b)s(e)36 b(the)h(h)m(yp)s(erplane)e(p)s(erp)s (endicular)f(to)j(the)g(segmen)m(t)f([)p Fm(x;)15 b(y)s Fn(])37 b(and)f(crossing)h(it)g(at)g(the)f(midp)s(oin)m(t)150 394 y(\()p Fm(x)8 b Fn(+)g Fm(y)s Fn(\))p Fm(=)p Fn(2.)41 b(Note)25 b(that)g Fm(y)j Fn(is)c(necessarily)h(closer)h(to)f Fm(x)1977 408 y Fj(0)2041 394 y Fn(then)f Fm(x)g Fn(is,)i(b)s(ecause)f Fm(C)31 b Fn(is)24 b(a)h(cap)g(cen)m(tered)g(at)g Fm(x)3685 408 y Fj(0)3725 394 y Fn(,)150 506 y(and)32 b(the)h(neigh)m(b)s(orho)s (o)s(d)e(of)h Fm(y)k Fn(is)c(mostly)g(in)h Fm(C)39 b Fn(whereas)32 b(the)h(neigh)m(b)s(orho)s(o)s(d)e(of)h Fm(x)h Fn(is)f(mostly)g(not)h(in)150 619 y Fm(C)7 b Fn(.)40 b(Let)28 b(us)g(denote)h(b)m(y)g Fm(S)1034 633 y Fj(+)1121 619 y Fn(the)f(half)h(sphere)f(con)m(taining)i Fm(x)2227 633 y Fj(0)2266 619 y Fn(.)40 b(Then)27 b Fm(x)f Fl(2)f Fm(S)2786 633 y Fi(\000)2873 619 y Fn(and)j Fm(y)g Fl(2)d Fm(S)3263 633 y Fj(+)3321 619 y Fn(.)40 b(Applying)150 732 y(the)27 b(symmetrization)f Fm(B)j Fl(!)d Fm(B)1233 699 y Fi(\003)1298 732 y Fn(with)g(resp)s(ect)h(to)g(this)g(h)m(yp)s (erplane,)f(a)h(large)h(fraction)f(of)g Fm(B)5 b Fn(\()p Fm(x;)15 b(r)s Fn(\))27 b(will)150 845 y(b)s(e)h(transferred)h(in)m(to) h Fm(B)5 b Fn(\()p Fm(y)s(;)15 b(r)s Fn(\))29 b(while)g(no)g(p)s(oin)m (t)g(of)g Fm(C)24 b Fl(\\)18 b Fm(B)33 b Fn(is)c(transferred)f(to)i(a)g (p)s(oin)m(t)f(whic)m(h)g(is)g(not)g(in)150 958 y Fm(C)41 b Fn(\(observ)m(e)36 b(that)f(the)g(symmetrization)f(of)h Fm(C)41 b Fn(is)35 b Fm(C)7 b Fn(\).)53 b(Th)m(us,)35 b Fm(\026)p Fn(\()p Fm(B)2551 925 y Fi(\003)2613 958 y Fl(\\)23 b Fm(C)7 b Fn(\))32 b Fm(>)g(\026)p Fn(\()p Fm(B)c Fl(\\)23 b Fm(C)7 b Fn(\).)53 b(Since)35 b Fm(B)3711 925 y Fi(\003)150 1071 y Fn(also)c(b)s(elongs)g(to)g Fl(B)h Fn(w)m(e)f(get)h(a)f(con)m(tradiction.)118 b Fa(2)291 1227 y Fg(Pro)s(of)37 b(of)g(Lemma)f(22:)80 b Fn(Note)33 b(that)f(the)g(function)f Fm(\026)p Fn(\()p Fl(\001)p Fn(\))h(is)g(not)g(con)m(tin)m(uous)g(on)g Fl(C)5 b Fn(.)44 b(\(Consider)150 1340 y(for)33 b(example)g(a)g(sequence)h(of)f (\014nite)g(sets)g Fl(f)p Fm(A)1743 1354 y Fk(n)1791 1340 y Fl(g)1836 1307 y Fi(1)1836 1363 y Fk(n)p Fj(=1)2007 1340 y Fn(suc)m(h)f(that)i Fm(A)2482 1354 y Fk(n)2559 1340 y Fl(!)c Fm(S)2741 1307 y Fk(n)p Fi(\000)p Fj(1)2878 1340 y Fn(.)49 b(Then)32 b Fm(\026)p Fn(\()p Fm(A)3350 1354 y Fk(n)3397 1340 y Fn(\))e(=)g(0)j(for)150 1453 y(all)g Fm(n)f Fn(but)g Fm(\026)p Fn(\()p Fm(S)685 1420 y Fk(n)p Fi(\000)p Fj(1)822 1453 y Fn(\))d(=)f(1.\))48 b(This)32 b(sho)m(ws,)g(for)h(example,)f(that)h(the)g(\014rst)f (condition)h(in)f(the)g(de\014nition)150 1566 y(of)e Fl(B)i Fn(is)e(not)g(a)g(closed)h(condition)f(and)f(explains)h(the)g (reason)g(for)f(the)h(somewhat)f(cum)m(b)s(ersome)f(pro)s(of)150 1679 y(b)s(elo)m(w.)41 b(What)30 b(sa)m(v)m(es)i(the)e(da)m(y)g(is)g (the)h(easily)g(established)f(fact)h(that,)g(in)e(spite)i(of)f(this)g (discon)m(tin)m(uit)m(y)-8 b(,)150 1792 y(for)30 b(all)h(closed)h Fm(B)5 b Fn(,)30 b Fm(\026)p Fn(\()p Fm(B)972 1806 y Fk(\017)1004 1792 y Fn(\))c Fl(!)f Fm(\026)p Fn(\()p Fm(B)5 b Fn(\).)291 1905 y(Let)38 b Fm(B)530 1919 y Fk(n)614 1905 y Fn(b)s(e)e(a)i(sequence)f(in)g Fl(B)j Fn(tending)d(to)h Fm(B)j Fn(in)c(the)g(Hausdor\013)g(metric.)61 b(F)-8 b(or)38 b(ev)m(ery)g Fm(\017;)15 b(\016)40 b(>)d Fn(0)150 2018 y(\()p Fm(B)254 2032 y Fk(n)301 2018 y Fn(\))336 2033 y Fk(\017)p Fj(+)p Fk(\016)495 2018 y Fl(\023)g Fm(B)672 2032 y Fk(\017)742 2018 y Fn(if)h Fm(n)f Fn(is)g(large)i (enough)e(and)g Fm(\026)p Fn(\(\()p Fm(B)1951 2032 y Fk(n)1999 2018 y Fn(\))2034 2033 y Fk(\017)p Fj(+)p Fk(\016)2155 2018 y Fn(\))h Fl(\024)f Fm(\026)p Fn(\()p Fm(A)2494 2033 y Fk(\017)p Fj(+)p Fk(\016)2616 2018 y Fn(\).)62 b(Sending)37 b Fm(\016)k Fn(to)d(zero)h(w)m(e)f(get)150 2130 y Fm(\026)p Fn(\()p Fm(B)309 2144 y Fk(\017)342 2130 y Fn(\))25 b Fl(\024)g Fm(\026)p Fn(\()p Fm(A)656 2144 y Fk(\017)689 2130 y Fn(\).)39 b(This)24 b(sho)m(ws)g(that)i Fm(B)j Fn(satis\014es)c(the)f(last)i(requiremen)m(t)d(and)h(also)i (\(sending)e Fm(\017)g Fn(to)i(zero\))150 2243 y(that)34 b Fm(\026)p Fn(\()p Fm(B)5 b Fn(\))31 b Fl(\024)f Fm(\026)p Fn(\()p Fm(A)p Fn(\).)51 b(That)34 b Fm(\026)p Fn(\()p Fm(B)5 b Fn(\))30 b Fl(\025)h Fm(\026)p Fn(\()p Fm(A)p Fn(\))j(follo)m(ws)h(from)d Fm(\026)p Fn(\()p Fm(B)2418 2257 y Fk(\017)2450 2243 y Fn(\))f Fl(\025)g Fm(\026)p Fn(\()p Fm(B)2777 2257 y Fk(n)2824 2243 y Fn(\))g(=)f Fm(\026)p Fn(\()p Fm(A)p Fn(\))k(for)g(an)m(y)g Fm(\017)c(>)h Fn(0)150 2356 y(pro)m(vided)h Fm(n)f Fn(is)h(large)h(enough.)45 b(This)31 b(sho)m(ws)h(that)h Fm(B)j Fn(satis\014ed)c(the)g(\014rst)f (requiremen)m(t)h(of)g(b)s(elonging)150 2469 y(to)39 b Fl(B)s Fn(.)64 b(T)-8 b(o)39 b(sho)m(w)f(that)h(it)g(satis\014es)g (the)f(second)g(requiremen)m(t,)i(note)f(\014rst)f(that,)j(since)e Fm(\026)p Fn(\()p Fm(B)3477 2483 y Fj(2)p Fk(\017)3570 2469 y Fl(n)26 b Fm(B)5 b Fn(\))150 2582 y(tends)30 b(to)h(zero)g(as)g Fm(\017)f Fn(tends)g(to)h(zero,)375 2767 y Fm(a)p Fn(\()p Fm(\017)p Fn(\))26 b(=)f Fm(\026)707 2729 y Fj(2)747 2767 y Fn(\()p Fl(f)p Fn(\()p Fm(x;)15 b(y)s Fn(\);)47 b Fm(x)25 b Fl(2)g Fm(B)1341 2781 y Fj(2)p Fk(\017)1409 2767 y Fm(;)46 b(y)28 b Fl(62)d Fm(B)5 b(;)45 b Fn(the)30 b(distance)h(b)s(et)m(w)m(een)g Fm(x)g Fn(and)e Fm(y)34 b Fn(is)c(at)h(least)h Fm(\032)p Fl(g)p Fn(\))150 2951 y(tends,)43 b(as)d Fm(\017)h Fn(tends)e(to)j(zero,)i(to)d Fm(\026)1410 2965 y Fk(\032)1450 2951 y Fn(\()p Fm(B)5 b Fn(\).)71 b(On)39 b(the)i(other)f(hand,)i(if)f Fm(n)f Fn(is)g(large)h(enough)g(to)g(ensure)150 3064 y Fm(B)30 b Fl(\022)25 b Fn(\()p Fm(B)449 3078 y Fk(n)496 3064 y Fn(\))531 3078 y Fk(\017)589 3064 y Fl(\022)g Fm(B)754 3078 y Fj(2)p Fk(\017)822 3064 y Fn(,)j(then)f Fm(a)p Fn(\()p Fm(\017)p Fn(\))f Fl(\025)f Fm(\026)1411 3078 y Fk(\032)1451 3064 y Fn(\(\()p Fm(B)1590 3078 y Fk(n)1638 3064 y Fn(\))1673 3078 y Fk(\017)1706 3064 y Fn(\))j(.)39 b(Finally)-8 b(,)30 b(since)e Fm(\026)p Fn(\(\()p Fm(B)2572 3078 y Fk(n)2619 3064 y Fn(\))2654 3078 y Fk(\017)2701 3064 y Fl(n)14 b Fm(B)2829 3078 y Fk(n)2877 3064 y Fn(\))26 b Fl(\024)f Fm(\026)p Fn(\()p Fm(A)3192 3078 y Fk(\017)3239 3064 y Fl(n)14 b Fm(A)p Fn(\))28 b(and)f(this)150 3177 y(tends)k(to)i(zero)g(as)f Fm(\017)g Fn(tends)f(to)i(zero,)g(w)m(e)g (get)g(that)f Fm(\026)2001 3191 y Fk(\032)2041 3177 y Fn(\(\()p Fm(B)2180 3191 y Fk(n)2228 3177 y Fn(\))2263 3191 y Fk(\017)2296 3177 y Fn(\))g(tends,)g(as)g Fm(\017)g Fn(tend)g(to)g(zero,)i(to)e Fm(\026)3523 3191 y Fk(\032)3563 3177 y Fn(\()p Fm(B)3667 3191 y Fk(n)3715 3177 y Fn(\))150 3290 y Ff(uniformly)g Fn(in)f Fm(n)p Fn(.)42 b(So,)31 b(giv)m(en)h(a)g Fm(\016)e(>)c Fn(0)31 b(\014nd)f(an)h Fm(\017)26 b(>)g Fn(0)31 b(suc)m(h)g(that)h Fm(\026)2507 3304 y Fk(\032)2547 3290 y Fn(\(\()p Fm(B)2686 3304 y Fk(n)2733 3290 y Fn(\))2768 3304 y Fk(\017)2802 3290 y Fn(\))26 b Fl(\025)g Fm(\026)3015 3304 y Fk(\032)3055 3290 y Fn(\()p Fm(B)3159 3304 y Fk(n)3207 3290 y Fn(\))21 b Fl(\000)f Fm(\016)35 b Fn(for)c(all)g Fm(n)150 3403 y Fn(and)f(also)h Fm(\026)564 3417 y Fk(\032)604 3403 y Fn(\()p Fm(B)5 b Fn(\))26 b Fl(\025)f Fm(a)p Fn(\()p Fm(\017)p Fn(\))20 b Fl(\000)g Fm(\016)s Fn(.)42 b(Then)29 b(if)i Fm(n)e Fn(is)i(large)g(enough,)694 3588 y Fm(\026)749 3602 y Fk(\032)789 3588 y Fn(\()p Fm(B)5 b Fn(\))26 b Fl(\025)f Fm(a)p Fn(\()p Fm(\017)p Fn(\))c Fl(\000)f Fm(\016)29 b Fl(\025)c Fm(\026)1542 3602 y Fk(\032)1582 3588 y Fn(\(\()p Fm(B)1721 3602 y Fk(n)1768 3588 y Fn(\))1803 3602 y Fk(\017)1836 3588 y Fn(\))c Fl(\000)f Fm(\016)29 b Fl(\025)c Fm(\026)2203 3602 y Fk(\032)2243 3588 y Fn(\()p Fm(B)2347 3602 y Fk(n)2394 3588 y Fn(\))c Fl(\000)f Fn(2)p Fm(\016)29 b Fl(\025)c Fm(\026)2806 3602 y Fk(\032)2846 3588 y Fn(\()p Fm(A)p Fn(\))c Fl(\000)f Fn(2)p Fm(\016)n(:)150 3772 y Fn(Since)30 b(this)h(holds)f(for)g(an)m(y)g Fm(\016)f(>)c Fn(0)31 b(w)m(e)g(get)h(that)f Fm(B)j Fn(satis\014es)d(the)g(second)f (requiremen)m(t)g(as)g(w)m(ell.)118 b Fa(2)150 4013 y Fb(6.2)112 b(The)38 b(sparse)g(v)m(ersion)150 4184 y Fn(W)-8 b(e)30 b(shall)e(use)h(standard)e(b)s(ounds)f(on)j(large)g (deviations)h(of)f(random)d(v)-5 b(ariables.)41 b(One)28 b(b)s(ound)e(that)j(w)m(e)150 4297 y(use)h(is)g(Cheb)m(ysc)m(hev's)h (inequalit)m(y)-8 b(.)150 4487 y Fg(Lemma)35 b(23)46 b Ff(F)-7 b(or)26 b(a)g(r)-5 b(andom)28 b(variable)e Fm(X)7 b Ff(,)27 b(let)f Fm(\026)f Fn(=)g Fm(E)5 b Fn([)p Fm(X)i Fn(])27 b Ff(denote)f(its)g(exp)-5 b(e)g(ctation,)29 b(let)c Fm(\033)3298 4454 y Fj(2)3363 4487 y Fn(=)g Fm(E)5 b Fn([\()p Fm(X)12 b Fl(\000)150 4600 y Fm(E)5 b Fn([)p Fm(X)i Fn(]\))389 4567 y Fj(2)430 4600 y Fn(])40 b Ff(denote)g(its)g (varianc)-5 b(e,)41 b(and)g(let)e Fm(\033)k Ff(denote)d(the)g(standar) -5 b(d)42 b(deviation)f(\(the)f(p)-5 b(ositive)40 b(squar)-5 b(e)150 4713 y(r)g(o)g(ot)35 b(of)e Fm(\033)497 4680 y Fj(2)536 4713 y Ff(\).)42 b(Then)33 b(for)g(every)g Fm(\025)25 b(>)g Fn(0)p Ff(,)1496 4936 y Fm(P)13 b Fn([)p Fl(j)p Fm(X)28 b Fl(\000)20 b Fm(\026)p Fl(j)25 b(\025)g Fm(\025\033)s Fn(])h Fl(\024)2300 4874 y Fn(1)p 2277 4914 93 4 v 2277 4998 a Fm(\025)2330 4972 y Fj(2)2379 4936 y Fm(:)1905 5247 y Fn(31)p eop %%Page: 32 32 32 31 bop 291 -58 a Fn(The)41 b(other)g(b)s(ound)f(that)h(w)m(e)h(use)f (deals)h(with)f(indep)s(enden)m(t)f(random)f(v)-5 b(ariables.)74 b(It)42 b(is)f(tak)m(en)150 55 y(from)29 b([15)q(],)i(and)f(the)g(pro)s (of)g(is)g(based)g(on)h([11)q(])f(and)g(is)g(omitted.)150 267 y Fg(Lemma)35 b(24)46 b Ff(L)-5 b(et)34 b Fm(X)904 281 y Fj(1)944 267 y Fm(;)15 b(:)g(:)g(:)h(;)f(X)1220 281 y Fk(n)1302 267 y Ff(b)-5 b(e)34 b(indep)-5 b(endent)35 b(r)-5 b(andom)36 b(variables,)g(with)e Fn(0)28 b Fl(\024)g Fm(X)3105 281 y Fk(i)3161 267 y Fl(\024)f Fn(1)35 b Ff(for)f(e)-5 b(ach)35 b Fm(i)p Ff(,)150 380 y(let)304 357 y Fn(\026)278 380 y Fm(X)d Fn(=)25 b(\()516 316 y Fe(P)620 380 y Fm(X)695 394 y Fk(i)723 380 y Fn(\))p Fm(=n)33 b Ff(and)h(let)e Fm(\026)25 b Fn(=)g Fm(E)5 b Fn([)1495 357 y(\026)1468 380 y Fm(X)j Fn(])p Ff(.)42 b(Then)33 b(for)g(every)g Fn(0)26 b Fm(<)f(\017)g(<)g Fn(1)1435 599 y Fm(P)13 b Fn([)1558 576 y(\026)1531 599 y Fm(X)28 b Fl(\000)20 b Fm(\026)25 b Fl(\025)g Fm(\017\026)p Fn(])g Fl(\024)g Fm(e)2181 562 y Fi(\000)2246 535 y Fc(1)p 2246 547 31 4 v 2246 588 a(3)2287 562 y Fk(\017)2316 538 y Fc(2)2350 562 y Fk(n\026)2439 599 y Fm(:)291 812 y Fn(The)k(pro)s(of)g(of)g(the)h (follo)m(wing)h(prop)s(osition)e(illustrates)i(the)e(use)g(of)h(the)g (large)g(deviation)h(b)s(ounds.)150 925 y(F)-8 b(uture)30 b(pro)s(ofs)g(will)h(b)s(e)e(giv)m(en)j(in)e(less)g(detail.)150 1137 y Fg(Lemma)35 b(25)46 b Ff(L)-5 b(et)35 b Fm(G)f Ff(b)-5 b(e)35 b(a)g(gr)-5 b(aph)37 b(with)f Fm(n)e Ff(vertic)-5 b(es)35 b(and)h Fm(m)f Ff(e)-5 b(dges,)35 b(and)h(let)f Fm(G)g Ff(b)-5 b(e)34 b(ne)-5 b(arly)37 b(r)-5 b(e)g(gular)36 b(in)150 1250 y(the)e(sense)g(that)h(its)f(highest)g(de)-5 b(gr)g(e)g(e)35 b Fn(\001)e Ff(is)h(b)-5 b(ounde)g(d)35 b(by)f(some)g(absolute)h(c)-5 b(onstant)35 b(times)f(the)g(aver)-5 b(age)150 1363 y(de)g(gr)g(e)g(e.)45 b(Sele)-5 b(ct)33 b(e)-5 b(ach)34 b(vertex)g(of)f Fm(G)g Ff(into)g Fm(G)1665 1330 y Fi(0)1721 1363 y Ff(indep)-5 b(endently)35 b(with)g(pr)-5 b(ob)g(ability)35 b Fm(p)26 b(>)g Fn(1)p Fm(=)p Fn(\001)p Ff(.)44 b(L)-5 b(et)34 b Fm(n)3550 1330 y Fi(0)3606 1363 y Ff(and)150 1476 y Fm(m)230 1443 y Fi(0)292 1476 y Ff(denote)40 b(the)g(numb)-5 b(er)39 b(of)h(vertic)-5 b(es)39 b(and)h(e)-5 b(dges)40 b(in)f Fm(G)2143 1443 y Fi(0)2165 1476 y Ff(.)62 b(Then)39 b(the)h(exp)-5 b(e)g(ctation)41 b(of)f Fm(n)3301 1443 y Fi(0)3362 1476 y Ff(is)g Fm(pn)e Ff(and)150 1589 y(of)h Fm(m)343 1556 y Fi(0)405 1589 y Ff(is)f Fm(p)554 1556 y Fj(2)594 1589 y Fm(m)p Ff(.)60 b(The)39 b(pr)-5 b(ob)g(ability)41 b(that)f Fm(n)1645 1556 y Fi(0)1706 1589 y Ff(deviates)f(by)g(an)g Fm(\017)g Ff(fr)-5 b(action)40 b(fr)-5 b(om)40 b(its)f(exp)-5 b(e)g(ctation)41 b(is)d(at)150 1702 y(most)g Fm(e)417 1666 y Fi(\000)482 1639 y Fc(1)p 482 1651 V 482 1692 a(3)522 1666 y Fk(\017)551 1643 y Fc(2)585 1666 y Fk(pn)668 1702 y Ff(,)f(and)g(the)g(pr)-5 b(ob)g(ability)39 b(that)f Fm(m)1780 1669 y Fi(0)1839 1702 y Ff(deviates)f(by)g(an)g Fm(\017)f Ff(fr)-5 b(action)38 b(fr)-5 b(om)38 b(its)e(exp)-5 b(e)g(ctation)39 b(is)150 1815 y Fm(O)s Fn(\(1)p Fm(=\017)384 1782 y Fj(2)424 1815 y Fm(pn)p Fn(\))p Ff(.)291 2027 y Fg(Pro)s(of:)61 b Fn(Let)41 b Fm(X)882 2041 y Fk(i)950 2027 y Fn(b)s(e)f(the)g(random)e(v)-5 b(ariable)41 b(that)g(is)f(1)h(if)f(v)m(ertex)h Fm(i)f Fn(is)g(c)m(hosen)h(in)m(to)g Fm(G)3455 1994 y Fi(0)3518 2027 y Fn(and)f(0)150 2140 y(otherwise.)80 b(Then)42 b Fm(n)934 2107 y Fi(0)1003 2140 y Fn(=)1121 2076 y Fe(P)1208 2102 y Fk(n)1208 2163 y(i)p Fj(=1)1342 2140 y Fm(X)1417 2154 y Fk(i)1445 2140 y Fn(.)79 b(F)-8 b(or)44 b(ev)m(ery)g Fm(i)p Fn(,)j Fm(E)5 b Fn([)p Fm(X)2253 2154 y Fk(i)2282 2140 y Fn(])47 b(=)g Fm(p)p Fn(.)79 b(By)43 b(linearit)m(y)i(of)e(exp)s (ectation,)150 2253 y Fm(E)5 b Fn([)247 2189 y Fe(P)350 2253 y Fm(X)425 2267 y Fk(i)454 2253 y Fn(])27 b(=)f Fm(pn)p Fn(.)43 b(The)31 b(b)s(ound)e(on)i(the)h(probabilit)m(y)g(that) g Fm(n)2245 2220 y Fi(0)2299 2253 y Fn(deviates)g(b)m(y)f Fm(\017pn)g Fn(from)f(its)h(exp)s(ectation)150 2366 y(follo)m(ws)g (from)e(Lemma)g(24,)i(substituting)f Fm(\026)25 b Fn(=)g Fm(p)p Fn(.)291 2479 y(No)m(w)33 b(w)m(e)g(turn)e(to)i Fm(m)1033 2446 y Fi(0)1056 2479 y Fn(.)47 b(F)-8 b(or)33 b(an)f(edge)h(\()p Fm(i;)15 b(j)5 b Fn(\))35 b(in)d Fm(G)p Fn(,)g(let)h Fm(Y)2271 2493 y Fk(ij)2364 2479 y Fn(b)s(e)f(the)g (random)f(v)-5 b(ariable)33 b(that)g(is)f(1)h(if)150 2592 y(edge)c(\()p Fm(i;)15 b(j)5 b Fn(\))30 b(is)e(c)m(hosen)h(in)m (to)g Fm(G)1200 2559 y Fi(0)1250 2592 y Fn(and)f(0)g(otherwise.)41 b(Then)27 b Fm(m)2253 2559 y Fi(0)2301 2592 y Fn(=)2397 2528 y Fe(P)2485 2615 y Fj(\()p Fk(i;j)t Fj(\))2635 2592 y Fm(Y)2688 2606 y Fk(ij)2748 2592 y Fn(.)40 b(As)28 b(eac)m(h)i(edge)e(is)g(selected)150 2716 y(only)38 b(if)h(its)f(t)m(w) m(o)i(endp)s(oin)m(ts)d(are)i(selected,)j(w)m(e)d(ha)m(v)m(e)g Fm(E)5 b Fn([)p Fm(Y)2228 2730 y Fk(ij)2289 2716 y Fn(])39 b(=)f Fm(p)2508 2683 y Fj(2)2547 2716 y Fn(.)64 b(By)39 b(linearit)m(y)h(of)e(exp)s(ectation,)150 2828 y Fm(\026)46 b Fn(=)g Fm(E)5 b Fn([)p Fm(m)545 2795 y Fi(0)569 2828 y Fn(])46 b(=)g Fm(p)803 2795 y Fj(2)842 2828 y Fm(m)p Fn(.)79 b(Let)43 b(us)f(no)m(w)h(b)s(ound)e(the)i(v)-5 b(ariance)44 b(of)f Fm(m)2559 2795 y Fi(0)2582 2828 y Fn(.)79 b(F)-8 b(or)44 b(t)m(w)m(o)g(edges)f(in)g Fm(G)p Fn(,)j(their)150 2941 y(selection)32 b(in)m(to)g Fm(G)776 2908 y Fi(0)829 2941 y Fn(is)f(indep)s(enden)m(t,)f(unless)f(they)i (share)g(a)g(v)m(ertex.)42 b(Indep)s(enden)m(t)29 b(pairs)h(of)h(edges) g(do)150 3054 y(not)g(con)m(tribute)g(to)g(the)f(v)-5 b(ariance.)42 b(This)30 b(giv)m(es:)534 3259 y Fm(E)5 b Fn([\()p Fm(m)746 3221 y Fi(0)790 3259 y Fl(\000)20 b Fm(E)5 b Fn([)p Fm(m)1058 3221 y Fi(0)1082 3259 y Fn(]\))1142 3221 y Fj(2)1182 3259 y Fn(])25 b(=)g Fm(E)5 b Fn([\()1465 3178 y Fe(X)1460 3366 y Fj(\()p Fk(i;j)t Fj(\))1592 3259 y Fn(\()p Fm(Y)1680 3273 y Fk(ij)1761 3259 y Fl(\000)20 b Fm(p)1898 3221 y Fj(2)1937 3259 y Fn(\)\))2007 3221 y Fj(2)2047 3259 y Fn(])26 b(=)f Fm(E)5 b Fn([)2374 3178 y Fe(X)2291 3366 y Fj(\()p Fk(i;j)t Fj(\))p Fk(;)p Fj(\()p Fk(k)r(;l)q Fj(\))2576 3259 y Fn(\()p Fm(Y)2664 3273 y Fk(ij)2745 3259 y Fl(\000)20 b Fm(p)2882 3221 y Fj(2)2921 3259 y Fn(\)\()p Fm(Y)3044 3274 y Fk(k)r(l)3130 3259 y Fl(\000)g Fm(p)3267 3221 y Fj(2)3306 3259 y Fn(\)])182 3580 y(=)284 3499 y Fe(X)278 3687 y Fj(\()p Fk(i;j)t Fj(\))424 3580 y Fm(E)5 b Fn([\()p Fm(Y)609 3594 y Fk(ij)691 3580 y Fl(\000)19 b Fm(p)827 3542 y Fj(2)867 3580 y Fn(\))902 3542 y Fj(2)941 3580 y Fn(])i(+)1153 3499 y Fe(X)1078 3687 y Fj(\()p Fk(i;j)t Fj(\))p Fk(;)p Fj(\()p Fk(i;l)q Fj(\))1364 3580 y Fm(E)5 b Fn([\()p Fm(Y)1549 3594 y Fk(ij)1630 3580 y Fl(\000)20 b Fm(p)1767 3542 y Fj(2)1806 3580 y Fn(\)\()p Fm(Y)1929 3595 y Fk(il)2000 3580 y Fl(\000)g Fm(p)2137 3542 y Fj(2)2176 3580 y Fn(\)])h(+)f(0)26 b Fl(\024)f Fm(p)2561 3542 y Fj(2)2600 3580 y Fm(m)20 b Fn(+)g Fm(O)s Fn(\()p Fm(p)2944 3542 y Fj(3)2983 3580 y Fn(\001)p Fm(m)p Fn(\))25 b(=)g Fm(O)s Fn(\()p Fm(p)3448 3542 y Fj(3)3488 3580 y Fn(\001)3564 3542 y Fj(2)3603 3580 y Fm(n)p Fn(\))p Fm(:)150 3842 y Fn(\(The)h(last)h(equalit)m(y)g (holds)e(b)s(ecause)h(w)m(e)h(assumed)d(that)i Fm(p)f(>)g Fn(1)p Fm(=)p Fn(\001.\))41 b(Hence)26 b(for)g(the)g(random)e(v)-5 b(ariable)150 3955 y Fm(m)230 3922 y Fi(0)283 3955 y Fn(w)m(e)31 b(ha)m(v)m(e)h Fm(\033)c Fn(=)d Fm(O)s Fn(\()p Fm(p)p Fn(\001)1032 3899 y Fl(p)p 1108 3899 101 4 v 56 x Fm(pn)o Fn(\).)41 b(Comparing)29 b Fm(\033)34 b Fn(and)29 b Fm(\026)h Fn(w)m(e)h(get)h(that)1136 4153 y Fm(\033)p 1136 4193 56 4 v 1136 4277 a(\026)1227 4214 y Fl(\024)25 b Fm(O)s Fn(\()1440 4149 y Fm(p)p Fn(\001)1562 4093 y Fl(p)p 1637 4093 101 4 v 1637 4149 a Fm(pn)p 1440 4193 298 4 v 1506 4277 a(p)1552 4251 y Fj(2)1592 4277 y Fm(m)1748 4214 y Fn(\))g Fl(\024)g Fm(O)s Fn(\()2021 4093 y Fl(p)p 2097 4093 101 4 v 56 x Fm(pn)p 2021 4193 177 4 v 2059 4277 a(pn)2207 4214 y Fn(\))h(=)f Fm(O)s Fn(\(1)p Fm(=)2561 4154 y Fl(p)p 2638 4154 101 4 v 2638 4214 a Fm(pn)o Fn(\))150 4464 y(No)m(w)34 b(the)f(concen)m(tration)h(of)f Fm(m)1275 4431 y Fi(0)1331 4464 y Fn(around)f(its)h(exp)s(ectation)i(follo)m(ws)f (from)d(Cheb)m(ysc)m(hev's)i(inequalit)m(y)-8 b(.)150 4577 y Fa(2)291 4690 y Fn(Lemma)41 b(25)j(implies)e(that)h(with)f(high) h(probabilit)m(y)-8 b(,)47 b Fm(n)2279 4657 y Fi(0)2348 4690 y Fl(')e Fm(pn)d Fn(and)h Fm(m)2877 4657 y Fi(0)2946 4690 y Fl(')i Fm(p)3108 4657 y Fj(2)3147 4690 y Fm(m)p Fn(,)h(meaning)c(in)150 4803 y(particular)33 b(that)g(the)g(a)m(v)m (erage)i(degree)e(in)f Fm(G)1723 4770 y Fi(0)1778 4803 y Fn(is)h(\002\()p Fm(p)p Fn(\001\).)47 b(Prop)s(osition)32 b(26)i(sho)m(ws)e(that)h Fm(G)3356 4770 y Fi(0)3411 4803 y Fn(is)g(nearly)150 4916 y(regular,)e(a)g(prop)s(ert)m(y)e(that)i (signi\014can)m(tly)h(simpli\014es)d(the)h(pro)s(ofs)g(that)h(follo)m (w.)1905 5247 y(32)p eop %%Page: 33 33 33 32 bop 150 -58 a Fg(Prop)s(osition)34 b(26)46 b Ff(L)-5 b(et)45 b Fm(p)h Fn(=)g Fm(c)15 b Fn(log)s(\()p Fm(n=)p Fn(\001\))p Fm(=)p Fn(\001)45 b Ff(for)g(a)g(su\016ciently)f(lar)-5 b(ge)45 b Fm(c)i(>)f Fn(0)p Ff(.)78 b(Then)44 b(with)h(high)150 55 y(pr)-5 b(ob)g(ability,)35 b(the)e(maximum)h(de)-5 b(gr)g(e)g(e)33 b Fn(\001)1534 22 y Fi(0)1590 55 y Ff(of)g Fm(G)1769 22 y Fi(0)1824 55 y Ff(is)g(at)g(most)h Fn(2)p Fm(p)p Fn(\001)p Ff(.)291 267 y Fg(Pro)s(of:)44 b Fn(Consider)31 b(an)g(arbitrary)g(v)m(ertex)i Fm(v)d Fl(2)d Fm(G)p Fn(.)44 b(The)31 b(probabilit)m(y)h(that)g(it)g(is)g(selected)h(in)m(to)f Fm(G)3727 234 y Fi(0)150 380 y Fn(is)26 b Fm(p)p Fn(.)39 b(The)25 b(exp)s(ectation)i(of)f(the)g(n)m(um)m(b)s(er)e(of)h(its)i (neigh)m(b)s(ors)e(that)h(are)g(selected)i(in)m(to)e Fm(G)3143 347 y Fi(0)3192 380 y Fn(is)g(at)g(most)f Fm(p)p Fn(\001.)150 493 y(By)37 b(Lemma)f(24,)k(the)d(probabilit)m(y)h(that)g (the)f(n)m(um)m(b)s(er)e(of)i(neigh)m(b)s(ors)g(is)g(t)m(wice)i(as)e(m) m(uc)m(h)f(is)h Fm(e)3500 460 y Fi(\000)p Fk(p)p Fj(\001)p Fk(=)p Fj(3)3725 493 y Fn(.)150 606 y(Hence)27 b(the)e(probabilit)m(y)i (that)f Fm(v)j Fn(giv)m(es)e(a)f(v)m(ertex)h(in)f Fm(G)2027 573 y Fi(0)2075 606 y Fn(with)f(degree)i(ab)s(o)m(v)m(e)g(2)p Fm(p)p Fn(\001)e(is)h(at)h(most)e Fm(pe)3501 573 y Fi(\000)p Fk(p)p Fj(\001)p Fk(=)p Fj(3)3725 606 y Fn(.)150 719 y(Using)k Fm(p)c Fn(=)g Fm(c)15 b Fn(log)s(\()p Fm(n=)p Fn(\001\))p Fm(=)p Fn(\001)29 b(for)g(a)g(su\016cien)m(tly)h(large)g Fm(c)25 b(>)g Fn(0,)30 b(this)f(probabilit)m(y)g(is)g(m)m(uc)m(h)f (smaller)g(than)150 832 y(1)p Fm(=n)p Fn(.)49 b(Using)33 b(the)h(union)e(b)s(ound)f(o)m(v)m(er)j(all)g(v)m(ertices)h(of)e Fm(G)p Fn(,)g(no)g(v)m(ertex)h(of)f Fm(G)2804 799 y Fi(0)2860 832 y Fn(will)h(ha)m(v)m(e)g(large)g(degree.)150 945 y Fa(2)291 1058 y Fn(Summarizing)27 b(prop)s(erties)i(of)g Fm(G)1442 1025 y Fi(0)1494 1058 y Fn(that)h(w)m(e)g(established)g(in)f (Lemma)e(25)k(and)d(Prop)s(osition)i(26,)g(w)m(e)150 1171 y(ha)m(v)m(e:)150 1383 y Fg(Corollary)j(27)46 b Ff(L)-5 b(et)39 b Fm(\017)e(>)f Fn(0)j Ff(b)-5 b(e)39 b(an)g(arbitr)-5 b(arily)41 b(smal)5 b(l)40 b(c)-5 b(onstant)40 b(and)g(let)f Fm(c)g Ff(b)-5 b(e)39 b(a)g(su\016ciently)g(lar)-5 b(ge)150 1496 y(c)g(onstant.)41 b(Given)25 b(a)g(ne)-5 b(arly)27 b(r)-5 b(e)g(gular)26 b(gr)-5 b(aph)27 b Fm(G)d Ff(and)i Fm(p)f Fn(=)g Fm(c)15 b Fn(log)s(\()2333 1460 y Fk(n)p 2325 1475 59 4 v 2325 1527 a Fj(\001)2394 1496 y Fn(\))p Fm(=)p Fn(\001)p Ff(,)27 b(then)e(a)h(r)-5 b(andom)27 b(vertex)e(induc)-5 b(e)g(d)150 1609 y(sub)g(gr)g(aph)34 b Fm(G)592 1576 y Fi(0)646 1609 y Ff(that)f(c)-5 b(ontains)33 b(every)f(vertex)g(of)g Fm(G)f Ff(indep)-5 b(endently)34 b(with)e(pr)-5 b(ob)g(ability)35 b Fm(p)c Ff(is)h Fn(t)m(ypical)h Ff(with)150 1722 y(high)k(pr)-5 b(ob)g(ability.)56 b(Her)-5 b(e)36 b(typic)-5 b(al)38 b(me)-5 b(ans)37 b(that)h Fn(\(1)24 b Fl(\000)e Fm(\017)p Fn(\))p Fm(pn)32 b Fl(\024)g Fm(n)2381 1689 y Fi(0)2436 1722 y Fl(\024)g Fn(\(1)24 b(+)e Fm(\017)p Fn(\))p Fm(pn)p Ff(,)37 b Fn(\(1)24 b Fl(\000)f Fm(\017)p Fn(\))p Fm(p)3290 1689 y Fj(2)3329 1722 y Fm(m)32 b Fl(\024)g Fm(m)3624 1689 y Fi(0)3679 1722 y Fl(\024)150 1835 y Fn(\(1)21 b(+)f Fm(\017)p Fn(\))p Fm(p)460 1802 y Fj(2)499 1835 y Fm(m)p Ff(,)33 b(and)g Fn(\001)892 1802 y Fi(0)940 1835 y Fl(\024)25 b Fn(2)p Fm(p)p Fn(\001)p Ff(.)291 2097 y Fg(Pro)s(of)38 b(of)f(Lemma)f(8:)81 b Fn(W)-8 b(e)33 b(use)f(the)g(notation)h(\()p Fm(X)r(;)15 b(Y)21 b Fn(\))33 b(to)f(denote)h(the)f(set)h(of)f(edges)g(b)s(et)m(w)m(een) 150 2210 y(disjoin)m(t)d(sets)h Fm(X)36 b Fn(and)29 b Fm(Y)20 b Fn(,)29 b(and)f Fl(j)p Fn(\()p Fm(X)r(;)15 b(Y)22 b Fn(\))p Fl(j)29 b Fn(to)h(denote)g(the)f(n)m(um)m(b)s(er)e(of) i(these)g(edges.)41 b(T)m(ypically)-8 b(,)31 b Fm(X)3593 2146 y Fe(S)3677 2210 y Fm(Y)150 2323 y Fn(will)i(b)s(e)g(all)h(v)m (ertices)g(of)g(the)f(resp)s(ectiv)m(e)h(graph,)f(in)g(whic)m(h)g(case) h(w)m(e)f(sa)m(y)h(that)g(\()p Fm(X)r(;)15 b(Y)21 b Fn(\))33 b(is)g(a)h(cut.)48 b(Let)150 2436 y Fm(G)29 b Fn(ha)m(v)m(e)h(the)g(v)m (ertex)h(set)f Fm(V)49 b Fn(and)29 b(let)h(the)g(random)d(subgraph)h Fm(G)2393 2403 y Fi(0)2445 2436 y Fn(ha)m(v)m(e)j(the)e(v)m(ertex)i (set)f Fm(V)3298 2403 y Fi(0)3321 2436 y Fn(.)41 b(F)-8 b(or)30 b(a)g(cut)150 2549 y(\()p Fm(A;)15 b(B)5 b Fn(\))31 b(in)f Fm(G)p Fn(,)g(its)h Ff(pr)-5 b(oje)g(ction)32 b Fn(on)f Fm(G)1411 2516 y Fi(0)1464 2549 y Fn(is)f(the)h(cut)f(\()p Fm(A)1971 2516 y Fi(0)1995 2549 y Fm(;)15 b(B)2109 2516 y Fi(0)2132 2549 y Fn(\))31 b(where)f Fm(A)2529 2516 y Fi(0)2578 2549 y Fn(=)25 b Fm(A)2757 2485 y Fe(T)2841 2549 y Fm(V)2914 2516 y Fi(0)2968 2549 y Fn(and)30 b Fm(B)3219 2516 y Fi(0)3267 2549 y Fn(=)25 b Fm(B)3452 2485 y Fe(T)3536 2549 y Fm(V)3609 2516 y Fi(0)3632 2549 y Fn(.)291 2662 y(Consider)k(a)i(maxim)m(um)c(cut)j(\()p Fm(A;)15 b(B)5 b Fn(\))31 b(in)f Fm(G)f Fn(that)i(cuts)f(opt)2336 2684 y Fk(d)2396 2662 y Fl(\001)20 b Fm(m)30 b Fn(edges.)41 b(Then)29 b(the)h(pro)5 b(jected)31 b(cut)150 2775 y(\()p Fm(A)253 2742 y Fi(0)277 2775 y Fm(;)15 b(B)391 2742 y Fi(0)414 2775 y Fn(\))34 b(is)g(exp)s(ected)g(to)h(cut)f Fm(p)1281 2742 y Fj(2)1320 2775 y Fn(opt)1451 2796 y Fk(d)1514 2775 y Fl(\001)23 b Fm(m)33 b Fn(in)h Fm(G)1857 2742 y Fi(0)1879 2775 y Fn(.)51 b(By)34 b(Cheb)m(ysc)m(hev's)g (inequalit)m(y)-8 b(,)37 b(with)c(probabilit)m(y)150 2888 y(1)13 b Fl(\000)g Fm(o)p Fn(\(1\),)29 b Fl(j)p Fn(\()p Fm(A)633 2855 y Fi(0)657 2888 y Fm(;)15 b(B)771 2855 y Fi(0)794 2888 y Fn(\))p Fl(j)27 b Fn(do)s(es)g(not)g(deviate)h (from)d(the)i(exp)s(ectation)h(b)m(y)f(more)e(than)i Fm(o)p Fn(\()p Fm(p)3083 2855 y Fj(2)3122 2888 y Fn(opt)3254 2909 y Fk(d)3307 2888 y Fl(\001)13 b Fm(m)p Fn(\).)40 b(Using)150 3000 y(Lemma)25 b(25)k(w)m(e)e(get)i(that)f Fm(m)1143 2967 y Fi(0)1191 3000 y Fl(')d Fm(p)1333 2967 y Fj(2)1372 3000 y Fm(m)p Fn(.)40 b(Hence)28 b(\()p Fm(A)1887 2967 y Fi(0)1911 3000 y Fm(;)15 b(B)2025 2967 y Fi(0)2048 3000 y Fn(\))28 b(cuts)f(roughly)g(an)g(opt)2883 3022 y Fk(d)2950 3000 y Fn(fraction)h(of)g(the)f(edges)150 3113 y(of)f Fm(G)321 3080 y Fi(0)344 3113 y Fn(.)39 b(As)26 b(the)h(maxim)m(um)c(cut)j(of)g Fm(G)1434 3080 y Fi(0)1483 3113 y Fn(cuts)h(at)f(least)i(as)e(m)m(uc)m(h,)h(w)m(e)f(ha)m(v)m(e)i (sho)m(wn)d(that)i(opt)3289 3135 y Fk(s)3351 3113 y Fl(\025)e Fn(opt)3578 3135 y Fk(d)3631 3113 y Fl(\000)12 b Fm(\017)150 3226 y Fn(with)30 b(high)g(probabilit)m(y)-8 b(.)291 3339 y(The)25 b(hard)f(part)h(of)g(the)g(Lemma)e(is)i(to)h(sho)m(w)f (that)h(opt)2175 3361 y Fk(s)2237 3339 y Fl(\024)f Fn(opt)2464 3361 y Fk(d)2514 3339 y Fn(+)10 b Fm(\017)p Fn(.)39 b(Eac)m(h)25 b(cut)g(in)g Fm(G)g Fn(cuts)g(at)h(most)150 3452 y(opt)281 3474 y Fk(d)340 3452 y Fl(\001)19 b Fm(m)29 b Fn(edges.)41 b(As)30 b(ab)s(o)m(v)m(e,)h(w)m(e)f(can)g(sho)m(w)f(that)h(its)g(pro)5 b(jection)30 b(on)g Fm(G)2661 3419 y Fi(0)2713 3452 y Fn(is)f(exp)s(ected)h(to)h(cut)e(at)i(most)150 3565 y(roughly)g(opt)613 3587 y Fk(d)675 3565 y Fl(\001)21 b Fm(m)801 3532 y Fi(0)855 3565 y Fn(edges.)45 b(Ho)m(w)m(ev)m(er,)34 b(there)d(is)h(some)f (probabilit)m(y)h(that)g(the)f(pro)5 b(jection)32 b(cuts)g(man)m(y)150 3678 y(more)27 b(edges.)40 b(The)27 b(probabilit)m(y)h(of)g(a)g(large)h (deviation)f(\(as)h(computed)d(b)m(y)h(Cheb)m(ysc)m(hev's)h(inequalit)m (y\))150 3791 y(is)i(small,)g(but)g(still)h(m)m(uc)m(h)e(larger)i(than) e(2)1604 3758 y Fi(\000)p Fk(n)1707 3791 y Fn(.)40 b(Because)32 b(the)e(n)m(um)m(b)s(er)e(of)i(cuts)g(in)g Fm(G)g Fn(is)g(2)3246 3758 y Fk(n)3293 3791 y Fn(,)h(w)m(e)f(cannot)150 3904 y(use)j(the)g(union)f(b)s(ound)f(to)i(argue)h(that)f(none)g(of)g(the)g (cuts)g(in)g Fm(G)f Fn(induces)h(an)g(abnormally)f(large)i(cut)150 4017 y(in)c Fm(G)328 3984 y Fi(0)351 4017 y Fn(.)291 4130 y(T)-8 b(o)31 b(o)m(v)m(ercome)h(this)f(problem)e(w)m(e)i(do)g(t)m (w)m(o)h(things.)41 b(One)30 b(is)h(to)g(sho)m(w)g(that)g(it)g (su\016ces)g(to)g(consider)150 4242 y(pro)5 b(jections)32 b(of)g(only)g(a)g(small)f(n)m(um)m(b)s(er)f(of)h Ff(sp)-5 b(e)g(cial)36 b(cuts)31 b Fn(in)h Fm(G)p Fn(,)f(rather)h(than)f(all)i (p)s(ossible)e(cuts)h(in)f Fm(G)p Fn(.)150 4355 y(The)k(other)h(is)f (for)g(eac)m(h)i(sp)s(ecial)f(cut)f(in)g Fm(G)p Fn(,)h(to)h(more)d (carefully)i(analyse)g(the)g(probabilit)m(y)g(that)g(the)150 4468 y(induced)g(cut)h(in)g Fm(G)843 4435 y Fi(0)903 4468 y Fn(is)g(abnormally)f(large,)k(and)d(pro)m(v)m(e)h(b)s(ounds)c (stronger)k(than)f(those)g(implied)f(b)m(y)150 4581 y(Cheb)m(ysc)m (hev's)31 b(inequalit)m(y)-8 b(.)150 4794 y Fg(De\014nition)33 b(4)46 b Ff(F)-7 b(or)34 b Fn(0)27 b Fl(\024)f Fm(\017)h Fl(\024)f Fn(1)p Ff(,)34 b(a)f(cut)g(in)g(a)h(gr)-5 b(aph)35 b(is)e Fm(\017)p Fn(-maxim)m(um)e Ff(if)i(the)g(numb)-5 b(er)34 b(of)g(e)-5 b(dges)33 b(in)h(the)150 4907 y(cut)e(is)h(at)g(le) -5 b(ast)34 b Fn(\(1)21 b Fl(\000)f Fm(\017)p Fn(\))33 b Ff(times)f(the)i(numb)-5 b(er)33 b(of)f(e)-5 b(dges)33 b(in)g(the)g(maximum)h(cut)e(in)h(the)g(gr)-5 b(aph.)1905 5247 y Fn(33)p eop %%Page: 34 34 34 33 bop 150 -58 a Fg(Lemma)35 b(28)46 b Ff(Ther)-5 b(e)34 b(is)g(a)h(c)-5 b(ol)5 b(le)-5 b(ction)35 b(of)f Fn(2)1669 -91 y Fk(O)r Fj(\()p Fk(n=\017)1859 -115 y Fc(4)1894 -91 y Fj(\001\))2018 -58 y Fn(sp)s(ecial)f(cuts)h Ff(in)g Fm(G)p Ff(,)f(such)h(that)i(with)f(high)f(pr)-5 b(ob-)150 55 y(ability)39 b(over)h(the)f(choic)-5 b(e)39 b(of)h Fm(G)1261 22 y Fi(0)1283 55 y Ff(,)h(the)e(pr)-5 b(oje)g(ction)41 b(of)e(one)g(of)g(these)h(cuts)f(is)f(an)i Fm(\017)p Ff(-maximum)f(cut)g(in)150 168 y Fm(G)222 135 y Fi(0)245 168 y Ff(.)291 380 y Fn(In)26 b([9)q(],)i(a)f(lemma)f (similar)g(to)i(Lemma)d(28)j(is)f(pro)m(v)m(ed)g(for)f(the)i(sp)s (ecial)f(case)h(that)g Fm(m)d Fn(=)g(\012\()p Fm(n)3455 347 y Fj(2)3494 380 y Fn(\).)39 b(The)150 493 y(pro)s(of)27 b(of)i([9])g(can)f(b)s(e)g(adapted)g(to)h(apply)e(also)i(to)g(lemma)e (28.)41 b(W)-8 b(e)29 b(assume)e(Lemma)f(28)j(and)f(con)m(tin)m(ue)150 606 y(with)i(the)h(pro)s(of)e(of)i(Lemma)d(8.)41 b(The)30 b(pro)s(of)g(of)g(Lemma)f(28)i(is)g(sk)m(etc)m(hed)g(later,)h(for)e (completeness.)291 719 y(In)m(tuitiv)m(ely)-8 b(,)33 b(w)m(e)e(w)m(ould)f(lik)m(e)h(to)h(no)m(w)e(pro)m(v)m(e)h(the)g(follo) m(wing:)286 932 y Fl(\017)46 b Fn(Consider)40 b(an)g(arbitrary)g(cut)h (\()p Fm(A;)15 b(B)5 b Fn(\))41 b(in)f Fm(G)p Fn(,)i(and)e(recall)i (that)f Fl(j)p Fn(\()p Fm(A;)15 b(B)5 b Fn(\))p Fl(j)43 b(\024)e Fn(opt)3251 953 y Fk(d)3318 932 y Fl(\001)27 b Fm(m)p Fn(.)70 b(With)377 1044 y(probabilit)m(y)31 b(2)887 1011 y Fi(\000)p Fj(\012\()p Fk(pn)p Fj(\))1161 1044 y Fn(o)m(v)m(er)h(the)e(c)m(hoice)j(of)d Fm(G)1958 1011 y Fi(0)1981 1044 y Fn(,)g Fl(j)p Fn(\()p Fm(A)2164 1011 y Fi(0)2189 1044 y Fm(;)15 b(B)2303 1011 y Fi(0)2326 1044 y Fn(\))p Fl(j)26 b(\025)f Fn(\(1)c(+)f Fm(\017)p Fn(\))p Fm(p)2818 1011 y Fj(2)2857 1044 y Fn(opt)2988 1066 y Fk(d)3049 1044 y Fl(\001)g Fm(m)p Fn(.)291 1257 y(When)34 b Fm(m)e Fn(=)f(\012\()p Fm(n)930 1224 y Fj(2)969 1257 y Fn(\),)36 b(this)e(follo)m(ws)h(from)e(a)i(pro)s(of)e(in)h([9)q (].)52 b(\(Note:)e(the)35 b(\012\()p Fm(pn)p Fn(\))e(notation)j(hides)e (a)150 1370 y(constan)m(t)39 b(that)f(dep)s(ends)e(on)i Fm(\017)p Fn(.\))63 b(But)37 b(in)h(our)f(case,)k(when)36 b Fm(m)i Fn(is)g(small,)h(this)e(migh)m(t)h(not)g(b)s(e)f(true.)150 1483 y(T)-8 b(o)30 b(o)m(v)m(ercome)h(this)f(problem,)e(w)m(e)j (condition)f(on)f Fm(G)1963 1450 y Fi(0)2016 1483 y Fn(b)s(eing)g(t)m (ypical.)42 b(Hence)31 b(w)m(e)f(pro)m(v)m(e)h(the)e(follo)m(wing)150 1596 y(mo)s(di\014ed)f(prop)s(ert)m(y)-8 b(.)150 1808 y Fg(Prop)s(osition)34 b(29)46 b Ff(F)-7 b(or)39 b(an)g(arbitr)-5 b(ary)40 b(cut)e Fn(\()p Fm(A;)15 b(B)5 b Fn(\))38 b Ff(in)g(a)g(ne)-5 b(arly)40 b(r)-5 b(e)g(gular)39 b(gr)-5 b(aph)40 b Fm(G)p Ff(,)e(the)h(pr)-5 b(ob)g(ability)150 1921 y(that)37 b Fl(j)p Fn(\()p Fm(A)467 1888 y Fi(0)491 1921 y Fm(;)15 b(B)605 1888 y Fi(0)628 1921 y Fn(\))p Fl(j)32 b(\025)e Fn(\(1)24 b(+)e Fm(\017)p Fn(\))p Fm(p)1136 1888 y Fj(2)1175 1921 y Ff(opt)1298 1943 y Fk(d)1361 1921 y Fl(\001)h Fm(m)35 b Ff(is)h(at)g(most)h Fn(2)2006 1888 y Fi(\000)p Fj(\012\()p Fk(\017)2168 1865 y Fc(2)2203 1888 y Fk(pn)p Fj(\))2313 1921 y Ff(,)f(wher)-5 b(e)37 b(pr)-5 b(ob)g(ability)38 b(is)e(c)-5 b(ompute)g(d)38 b(over)150 2034 y(the)33 b(choic)-5 b(e)33 b(of)g Fm(G)754 2001 y Fi(0)809 2034 y Fn(conditioned)e(on)f Fm(G)1497 2001 y Fi(0)1551 2034 y Fn(b)s(eing)f(t)m(ypical)j(in)f(the)f(sense)g (of)h(Corollary)g(27)p Ff(.)291 2247 y Fg(Pro)s(of:)52 b Fn(As)35 b Fm(G)836 2214 y Fi(0)894 2247 y Fn(is)h(assumed)d(to)j(b)s (e)f(t)m(ypical,)k(the)c(maxim)m(um)e(degree)j(in)f Fm(G)2997 2214 y Fi(0)3055 2247 y Fn(is)g(within)g(a)h(factor)150 2359 y(of)g(4)g(of)g(the)g(a)m(v)m(erage)i(degree.)58 b(This)35 b(implies)g(that)h(if)g Fl(j)p Fm(A)2188 2326 y Fi(0)2212 2359 y Fl(j)e Fm(<)g Fl(j)p Fm(B)2475 2326 y Fi(0)2498 2359 y Fl(j)p Fm(=)p Fn(16)k(then)d(\()p Fm(A)3011 2326 y Fi(0)3035 2359 y Fm(;)15 b(B)3149 2326 y Fi(0)3172 2359 y Fn(\))36 b(cuts)g(at)h(most)150 2472 y(1)p Fm(=)p Fn(2)48 b Fm(<)d Fn(opt)581 2494 y Fk(d)664 2472 y Fn(the)e(edges)h(of)f Fm(G)1277 2439 y Fi(0)1299 2472 y Fn(.)79 b(Hence)44 b(w)m(e)f(can)g(assume)f(that)h Fl(j)p Fm(A)2640 2439 y Fi(0)2664 2472 y Fl(j)j(\025)g(j)p Fm(B)2951 2439 y Fi(0)2974 2472 y Fl(j)p Fm(=)p Fn(16.)80 b(This)42 b(in)h(turn)150 2585 y(implies)34 b(that)h(w)m(e)g(can)f (assume)g(that)h Fl(j)p Fm(A)p Fl(j)d Fn(=)g(\012\()p Fm(n)p Fn(\),)k(b)s(ecause)e(otherwise,)i(with)f(probabilit)m(y)f(at)i (least)150 2698 y(1)21 b Fl(\000)f Fn(2)352 2665 y Fi(\000)p Fj(\012\()p Fk(pn)p Fj(\))595 2698 y Fn(,)31 b Fl(j)p Fm(A)744 2665 y Fi(0)768 2698 y Fl(j)25 b Fm(<)g Fl(j)p Fm(B)1013 2665 y Fi(0)1036 2698 y Fl(j)p Fm(=)p Fn(16)32 b(\(this)f(can)g(b)s(e)e(pro)m(v)m(ed)i(as)g(in)f(Lemma)e(25\).)42 b(Similarly)-8 b(,)31 b(w)m(e)f(can)h(assume)150 2811 y(that)g Fl(j)p Fm(B)5 b Fl(j)25 b Fn(=)g(\012\()p Fm(n)p Fn(\).)291 2924 y(T)-8 b(o)30 b(simplify)e(the)h(notation)h(in)f(the)h (argumen)m(ts)e(that)i(follo)m(w,)h(w)m(e)f(assume)e(that)i Fl(j)p Fn(\()p Fm(A;)15 b(B)5 b Fn(\))p Fl(j)26 b Fn(=)f(opt)3666 2946 y Fk(d)3725 2924 y Fl(\001)150 3037 y Fm(m)37 b Fn(=)f(\012\()p Fm(m)p Fn(\).)62 b(This)37 b(assumption)f(can)h(b)s(e)g (made)g(without)g(loss)h(of)f(generalit)m(y)-8 b(,)42 b(b)s(ecause)c(the)f(larger)150 3150 y Fl(j)p Fn(\()p Fm(A;)15 b(B)5 b Fn(\))p Fl(j)33 b Fn(is,)h(the)e(higher)g(the)g (probabilit)m(y)h(that)g Fl(j)p Fn(\()p Fm(A)1995 3117 y Fi(0)2019 3150 y Fm(;)15 b(B)2133 3117 y Fi(0)2156 3150 y Fn(\))p Fl(j)29 b(\025)f Fn(\(1)23 b(+)e Fm(\017)p Fn(\))p Fm(p)2657 3117 y Fj(2)2696 3150 y Fn(opt)2827 3172 y Fk(d)2889 3150 y Fl(\001)h Fm(m)p Fn(.)46 b(\(W)-8 b(e)34 b(w)m(aited)f(with)150 3263 y(this)i(assumption)e(un)m(til)h(w)m (e)i(established)e Fl(j)p Fm(A)p Fl(j)f Fn(=)g(\012\()p Fm(n)p Fn(\),)i(b)s(ecause)g(if)f Fl(j)p Fm(A)p Fl(j)g Fn(=)e Fm(o)p Fn(\()p Fm(n)p Fn(\))i(and)g(\001)f(=)f Fm(O)s Fn(\()p Fm(m=n)p Fn(\))150 3376 y(the)f(assumption)d(do)s(es)i (not)h(mak)m(e)f(sense.\))291 3489 y(No)m(w)k(consider)f(the)h (selection)h(of)e Fm(A)1564 3456 y Fi(0)1588 3489 y Fn(.)49 b(In)33 b(exp)s(ectation,)j Fl(j)p Fn(\()p Fm(A)2426 3456 y Fi(0)2450 3489 y Fm(;)15 b(B)5 b Fn(\))p Fl(j)30 b Fn(=)g Fm(p)p Fl(j)p Fn(\()p Fm(A;)15 b(B)5 b Fn(\))p Fl(j)p Fn(.)51 b(W)-8 b(e)34 b(claim)f(that)150 3601 y(the)27 b(probabilit)m(y)h(of)f(an)g Fm(\017)14 b Fl(\001)g Fm(p)g Fl(\001)g Fn(opt)1307 3623 y Fk(d)1361 3601 y Fl(\001)g Fm(m)27 b Fn(deviation)h(ab)s(o)m(v)m(e)g(the)g(exp)s (ectation)g(is)f(exp)s(onen)m(tially)i(small)d(in)150 3714 y Fm(\017)187 3681 y Fj(2)226 3714 y Fm(pn)p Fn(.)40 b(The)29 b(claim)g(is)h(pro)m(v)m(ed)g(using)f(Lemma)e(24.)42 b(T)-8 b(reat)30 b(eac)m(h)g(v)m(ertex)h Fm(v)i Fn(of)c Fm(A)h Fn(as)g(a)f(random)f(v)-5 b(ariable)150 3827 y(that)27 b(has)g(v)-5 b(alue)27 b Fl(j)p Fn(\()p Fm(v)s(;)15 b(B)5 b Fn(\))p Fl(j)p Fm(=)p Fn(\001)27 b Fl(\024)e Fn(1)i(if)g Fm(v)i Fl(2)c Fm(A)1637 3794 y Fi(0)1687 3827 y Fn(\(whic)m(h)i(happ)s (ens)e(with)i(probabilit)m(y)g Fm(p)p Fn(\))g(and)f(0)h(otherwise.)150 3940 y(Hence)40 b(in)e(the)h(terminology)g(of)g(Lemma)e(24,)42 b Fm(\026)e Fn(=)f Fm(p)p Fl(j)p Fn(\()p Fm(A;)15 b(B)5 b Fn(\))p Fl(j)p Fm(=)p Fn(\001)p Fl(j)p Fm(A)p Fl(j)41 b Fn(=)e(\002\()p Fm(p)p Fn(\).)66 b(Here)39 b(w)m(e)h(used)e(the)150 4053 y(assumptions)32 b(that)j Fl(j)p Fn(\()p Fm(A;)15 b(B)5 b Fn(\))p Fl(j)33 b Fn(=)e(\002\()p Fm(m)p Fn(\),)k Fl(j)p Fm(A)p Fl(j)d Fn(=)f(\002\()p Fm(n)p Fn(\))j(and)f(\001)f(=)f (\002\()p Fm(m=n)p Fn(\).)51 b(Lemma)33 b(24)h(implies)g(the)150 4166 y(claim.)291 4279 y(No)m(w)h(w)m(e)g(select)h(v)m(ertices)g(of)f Fm(B)k Fn(in)m(to)c Fm(B)1703 4246 y Fi(0)1726 4279 y Fn(.)53 b(In)34 b(exp)s(ectation,)j Fl(j)p Fn(\()p Fm(B)2576 4246 y Fi(0)2600 4279 y Fm(;)15 b(A)2708 4246 y Fi(0)2731 4279 y Fn(\))p Fl(j)33 b Fn(=)f Fm(p)p Fl(j)p Fn(\()p Fm(B)5 b(;)15 b(A)3215 4246 y Fi(0)3239 4279 y Fn(\))p Fl(j)p Fn(.)53 b(Again)35 b(w)m(e)150 4392 y(claim)29 b(that)h(the)f(probabilit)m(y)g(of)h(an)f Fm(\017)p Fn(-fraction)h (deviation)g(ab)s(o)m(v)m(e)g(the)f(exp)s(ectation)i(is)e(exp)s(onen)m (tially)150 4505 y(small)g(in)h Fm(\017)530 4472 y Fj(2)569 4505 y Fm(pn)p Fn(.)40 b(T)-8 b(o)31 b(pro)m(v)m(e)f(the)g(claim)g(w)m (e)h(use)e(the)h(conditioning)h(that)g Fm(G)2746 4472 y Fi(0)2798 4505 y Fn(is)f(t)m(ypical,)i(whic)m(h)e(implies)150 4618 y(that)39 b(the)f(largest)h(degree)g(of)f(a)g(v)m(ertex)h(from)e Fm(B)1885 4585 y Fi(0)1945 4618 y Fn(in)m(to)i Fm(A)2205 4585 y Fi(0)2267 4618 y Fn(is)f(at)g(most)g(2)p Fm(p)p Fn(\001.)63 b(View)38 b(eac)m(h)i(v)m(ertex)f Fm(v)150 4731 y Fn(of)k Fm(B)j Fn(as)d(a)g(random)d(v)-5 b(ariable)44 b(that)f(has)f(v)-5 b(alue)43 b Fl(j)p Fn(\()p Fm(v)s(;)15 b(A)2137 4698 y Fi(0)2161 4731 y Fn(\))p Fl(j)p Fm(=)p Fn(2)p Fm(p)p Fn(\001)47 b Fl(\024)e Fn(1)d(if)h Fm(v)48 b Fl(2)d Fm(B)3051 4698 y Fi(0)3116 4731 y Fn(\(whic)m(h)e(happ)s(ens) 150 4844 y(with)d(probabilit)m(y)g Fm(p)p Fn(\))g(and)f(0)h(otherwise.) 70 b(\(Note)41 b(that)g(b)m(y)e(our)h(conditioning,)j(if)d Fl(j)p Fn(\()p Fm(v)s(;)15 b(A)3345 4811 y Fi(0)3370 4844 y Fn(\))p Fl(j)41 b Fm(>)g Fn(2)p Fm(p)p Fn(\001)150 4956 y(then)d(necessarily)h Fm(v)i Fl(62)d Fm(B)1084 4923 y Fi(0)1107 4956 y Fn(,)i(implying)d(that)h(the)h(v)-5 b(alue)38 b(of)g(the)h(random)d(v)-5 b(ariable)39 b(asso)s(ciated)g (with)1905 5247 y(34)p eop %%Page: 35 35 35 34 bop 150 -58 a Fm(v)46 b Fn(is)d(indeed)f(nev)m(er)h(larger)g (than)g(1.\))78 b(In)42 b(the)h(terminology)g(of)g(Lemma)e(24,)46 b(w)m(e)e(ha)m(v)m(e)f(that)h Fm(\026)h Fl(\024)150 55 y Fm(p)p Fl(j)p Fn(\()p Fm(B)5 b(;)15 b(A)438 22 y Fi(0)462 55 y Fn(\))p Fl(j)p Fm(=)p Fn(2)p Fm(p)p Fn(\001)p Fl(j)p Fm(B)5 b Fl(j)26 b Fn(=)f(\002\()p Fm(p)p Fn(\).)41 b(Using)30 b(Lemma)f(24)i(the)g(claim)f(is)g(pro)m(v)m(ed.)291 168 y(The)k(t)m(w)m(o)h(claims)f(ab)s(o)m(v)m(e)i(imply)d(that)h (conditioned)h(on)f Fm(G)2356 135 y Fi(0)2413 168 y Fn(b)s(eing)g(t)m (ypical,)j(with)d(probabilit)m(y)g(at)150 281 y(most)c(2)418 248 y Fi(\000)p Fj(\012\()p Fk(\017)580 224 y Fc(2)615 248 y Fk(pn)p Fj(\))755 281 y Fn(w)m(e)h(ha)m(v)m(e)g Fl(j)p Fn(\()p Fm(A)1226 248 y Fi(0)1250 281 y Fm(;)15 b(B)1364 248 y Fi(0)1388 281 y Fn(\))p Fl(j)25 b Fm(>)g Fn(\(1)c(+)f Fm(\017)p Fn(\))p Fm(p)1879 248 y Fj(2)1919 281 y Fn(opt)2050 302 y Fk(d)2111 281 y Fl(\001)g Fm(m)p Fn(.)116 b Fa(2)291 394 y Fn(Let)31 b(us)f(summarize)e(the)j(prop)s (erties)e(that)i(w)m(e)g(ha)m(v)m(e)h(pro)m(v)m(ed)e(so)h(far.)286 581 y Fl(\017)46 b Fn(With)41 b(high)f(probabilit)m(y)h(o)m(v)m(er)h (the)f(c)m(hoice)h(of)f Fm(G)2147 548 y Fi(0)2170 581 y Fn(,)i Fm(G)2310 548 y Fi(0)2373 581 y Fn(is)e(t)m(ypical,)k(meaning) 39 b(that)j Fm(n)3447 548 y Fi(0)3512 581 y Fl(')f Fm(pn)p Fn(,)377 694 y Fm(m)457 661 y Fi(0)506 694 y Fl(')25 b Fm(p)648 661 y Fj(2)687 694 y Fm(m)p Fn(,)30 b(and)g(\001)1075 661 y Fi(0)1123 694 y Fm(<)25 b Fn(2)p Fm(p)p Fn(\001)30 b(\(Corollary)i(27\).)286 882 y Fl(\017)46 b Fn(There)41 b(are)h(2)859 849 y Fk(O)r Fj(\()p Fk(n=\017)1049 825 y Fc(4)1083 849 y Fj(\001\))1215 882 y Ff(sp)-5 b(e)g(cial)43 b Fn(cuts)e(in)g Fm(G)p Fn(.)73 b(With)41 b(high)g(probabilit)m(y)h(o)m (v)m(er)h(the)e(c)m(hoice)i(of)f(a)377 995 y(random)25 b(subgraph)f Fm(G)1169 962 y Fi(0)1192 995 y Fn(,)j(a)g(near)f(maximal) f(cut)h(in)g Fm(G)2207 962 y Fi(0)2256 995 y Fn(can)g(b)s(e)g(obtained) g(as)h(a)f(pro)5 b(jection)27 b(of)g(one)377 1108 y(of)k(the)f(sp)s (ecial)h(cuts)g(of)f Fm(G)g Fn(\(Lemma)f(28\).)286 1295 y Fl(\017)46 b Fn(F)-8 b(or)37 b(an)e(arbitrary)h(sp)s(ecial)g(cut)g (\()p Fm(A;)15 b(B)5 b Fn(\),)38 b(the)e(probabilit)m(y)g(o)m(v)m(er)h (the)f(c)m(hoice)h(of)f Fm(G)3298 1262 y Fi(0)3356 1295 y Fn(that)h Fm(G)3631 1262 y Fi(0)3689 1295 y Fn(is)377 1408 y(t)m(ypical)32 b(and)e Fl(j)p Fn(\()p Fm(A)980 1375 y Fi(0)1004 1408 y Fm(;)15 b(B)1118 1375 y Fi(0)1141 1408 y Fn(\))p Fl(j)26 b Fm(>)f Fn(\(1)c(+)f Fm(\017)p Fn(\))p Fm(p)1633 1375 y Fj(2)1672 1408 y Fn(opt)1804 1430 y Fk(d)1864 1408 y Fl(\001)h Fm(m)30 b Fn(b)s(oth)g(hold)g(is)g(2) 2573 1375 y Fi(\000)p Fj(\012\()p Fk(pn)p Fj(\))2817 1408 y Fn(.)291 1596 y(The)i(last)i(prop)s(ert)m(y)e(together)j(with)d 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2047 y Fn(+)21 b Fm(O)s Fn(\()p Fm(\017)p Fn(\).)46 b(This)150 2160 y(completes)31 b(the)f(pro)s(of)g(of)g(Lemma) f(8.)117 b Fa(2)291 2323 y Fg(Pro)s(of)39 b(of)f(Lemma)g(28:)46 b Fn(Let)33 b Fm(k)g Fn(=)d(\002\(1)p Fm(=\017)p Fn(\).)49 b(Consider)32 b(a)i(partition)f(of)h(the)f(v)m(ertices)h(of)g Fm(G)e Fn(in)m(to)150 2436 y Fm(k)j Fn(roughly)d(equal)h(size)g(sets)g Fm(V)1214 2450 y Fj(1)1269 2436 y Fm(:)15 b(:)g(:)h(V)1443 2451 y Fk(k)1486 2436 y Fn(,)33 b(where)f(the)g(n)m(um)m(b)s(er)e(of)j (edges)g(in)f(the)g(subgraph)f(induced)g(b)m(y)150 2549 y(eac)m(h)c(set)f(is)f Fm(O)s Fn(\()p Fm(\017)717 2516 y Fj(2)757 2549 y Fm(m)p Fn(\).)39 b(\(Lemma)24 b(25)j(implies)d(that)i (with)g(high)f(probabilit)m(y)-8 b(,)27 b(a)f(random)e(partition)i(on)g Fm(V)150 2662 y Fn(in)m(to)i Fm(k)h Fn(sets)e(will)g(result)g(in)f (subgraphs)f(with)h(roughly)g Fm(n=k)k Fn(v)m(ertices)e(and)e Fm(m=k)2908 2629 y Fj(2)2974 2662 y Fn(edges.\))41 b(In)25 b(addition,)150 2775 y(select)32 b(at)g(random)d(a)i(sequence)h Fm(S)j Fn(of)c(\002\()p Fm(k)1649 2742 y Fj(4)1689 2775 y Fm(n=)p Fn(\001\))g(distinct)g(v)m(ertices)i(from)c Fm(G)p Fn(,)i(and)f(partition)h(in)g(in)m(to)150 2888 y Fm(k)40 b Fn(equal)e(length)g(consecutiv)m(e)h(subsequences)e Fm(S)1869 2902 y Fj(1)1923 2888 y Fm(:)15 b(:)g(:)h(S)2100 2903 y Fk(k)2143 2888 y Fn(.)61 b(\(Hence)39 b Fl(j)p Fm(S)2623 2902 y Fk(i)2651 2888 y Fl(j)e Fn(=)g(\002\()p Fm(k)2977 2855 y Fj(3)3016 2888 y Fm(n=)p Fn(\001\))h(for)f(ev)m(ery)h Fm(i)p Fn(.\))150 3000 y(Consider)32 b(all)i(2)706 2967 y Fi(j)p Fk(S)t Fi(j)829 3000 y Fn(p)s(ossibilities)g(of)f(t)m(w)m (o-coloring)j(the)d(v)m(ertices)i(in)e Fm(S)38 b Fn(as)33 b Ff(left)g Fn(and)f Ff(right)p Fn(.)50 b(Eac)m(h)33 b(suc)m(h)150 3113 y(p)s(ossibilit)m(y)j(will)g(de\014ne)g(a)g(unique)f (cut)h(in)f Fm(G)g Fn(as)i(follo)m(ws.)58 b(A)36 b(v)m(ertex)h Fm(v)g Fl(2)d Fm(V)2878 3127 y Fk(i)2942 3113 y Fn(is)i(placed)g(on)g (the)g(left)150 3226 y(side)e(if)g(more)e(of)i(its)h(neigh)m(b)s(ors)e (from)f Fm(S)1578 3240 y Fk(i)1640 3226 y Fn(are)i(colored)h Ff(right)f Fn(than)g Ff(left)p Fn(.)51 b(Otherwise)34 b(it)g(is)g(placed)g(on)150 3339 y(the)i(righ)m(t)g(side.)55 b(W)-8 b(e)37 b(sa)m(y)f(that)g Fm(S)k Fn(is)c Ff(go)-5 b(o)g(d)37 b Fn(for)e Fm(G)g Fn(and)g Fm(V)2181 3353 y Fj(1)2220 3339 y Fm(;)15 b(:)g(:)g(:)i(;)e(V)2475 3354 y Fk(k)2553 3339 y Fn(if)36 b(at)g(least)g(one)g(of)g(the)f(2)3462 3306 y Fi(j)p Fk(S)t Fi(j)3588 3339 y Fn(cuts)150 3452 y(generated)c(ab)s(o)m(v)m(e)h(forms)d(an)h Fm(\017)p Fn(-maxim)m(um)d(cut.)150 3665 y Fg(Prop)s(osition)34 b(30)46 b Ff(F)-7 b(or)34 b Fm(G)e Ff(and)h Fm(V)1368 3679 y Fj(1)1408 3665 y Fm(;)15 b(:)g(:)g(:)i(;)e(V)1663 3680 y Fk(k)1738 3665 y Ff(as)33 b(ab)-5 b(ove,)33 b(almost)i(al)5 b(l)33 b(choic)-5 b(es)33 b(of)g Fm(S)38 b Ff(ar)-5 b(e)33 b(go)-5 b(o)g(d.)291 3877 y Fn(Let)40 b(us)e(\014rst)h(note)h(some)e (subtleties)i(in)m(v)m(olv)m(ed)h(in)e(the)h(pro)s(of)e(of)i(prop)s 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Fk(i)1309 4555 y Fn(are)h(colored)h Ff(left)e Fn(\(whic)m(h)g(is)h(consisten)m(t)g(with)f(the)h(true)f (maxim)m(um)150 4667 y(cut\))e(then)f(this)g(induces)g(a)h(v)m(ery)g (bad)f(cut)g(on)h Fm(G)p Fn(,)g(as)f(all)i(v)m(ertices)g(of)e(the)h (resp)s(ectiv)m(e)g Fm(V)3280 4681 y Fk(i)3343 4667 y Fn(end)f(up)f(on)150 4780 y(the)g(left)g(side.)46 b(T)-8 b(o)33 b(get)h(a)f(near)f(maxim)m(um)d(cut)k(of)f Fm(G)p Fn(,)h(w)m(e)g(will)f(need)g(to)i(t)m(w)m(o-color)h Fm(S)i Fn(in)32 b(a)h(w)m(a)m(y)g(that)150 4893 y(colors)e(some)f(of)h(its)f (clique)i(v)m(ertices)g Ff(left)p Fn(.)1905 5247 y(35)p eop %%Page: 36 36 36 35 bop 291 -58 a Fn(Prop)s(osition)22 b(30)g(is)g(pro)m(v)m(ed)g(in) g([9])g(for)g(the)g(case)h Fm(m)i Fn(=)g(\012\()p Fm(n)2289 -91 y Fj(2)2328 -58 y Fn(\))d(\(in)g(whic)m(h)f(case)i Fl(j)p Fm(S)5 b Fl(j)26 b Fn(=)e Fm(O)s Fn(\(1)p Fm(=\017)3419 -91 y Fj(4)3460 -58 y Fn(\)\).)38 b(The)150 55 y(pro)s(of)31 b(extends)g(to)h(our)f(case)i(w)m(ere)f Fm(m)f Fn(is)g(smaller,)h(b)s 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b(shall)e(sho)m(w)g(that)h(with)f(high)g(probabilit)m(y)-8 b(,)45 b(for)150 958 y(ev)m(ery)f(1)j Fl(\024)f Fm(i)h Fl(\024)f Fm(k)s Fn(,)g Fl(j)p Fn(\()p Fm(A)1056 972 y Fk(i)1085 958 y Fm(;)15 b(B)1194 972 y Fk(i)1223 958 y Fn(\))p Fl(j)47 b(\025)f(j)p Fn(\()p Fm(A)1575 972 y Fk(i)p Fi(\000)p Fj(1)1694 958 y Fm(;)15 b(B)1803 972 y Fk(i)p Fi(\000)p Fj(1)1922 958 y Fn(\))p Fl(j)29 b(\000)g Fm(\017)2148 925 y Fj(2)2187 958 y Fm(m)p Fn(.)79 b(As)43 b Fm(k)49 b Fn(=)d(\002\(1)p Fm(=\017)p Fn(\),)i(this)43 b(implies)f(that)150 1071 y Fl(j)p Fn(\()p Fm(A)278 1086 y Fk(k)321 1071 y Fm(;)15 b(B)430 1086 y Fk(k)473 1071 y Fn(\))p Fl(j)26 b(\025)f(j)p Fn(\()p Fm(A;)15 b(B)5 b Fn(\))p Fl(j)22 b(\000)e Fm(\017m)p Fn(,)30 b(establishing)h(that)g (\()p Fm(A)2038 1086 y Fk(k)2081 1071 y Fm(;)15 b(B)2190 1086 y Fk(k)2233 1071 y Fn(\))31 b(is)f(an)g Fm(\017)p Fn(-maxim)m(um)e(cut.)291 1184 y(The)j(n)m(um)m(b)s(er)e(of)j(edges)g (connecting)g(v)m(ertices)h(within)e(a)h(set)g Fm(V)2502 1198 y Fk(i)2561 1184 y Fn(is)g Fm(O)s Fn(\()p Fm(\017)2798 1151 y Fj(2)2837 1184 y Fm(m)p Fn(\),)g(and)f(the)h(n)m(um)m(b)s(er)d (of)150 1297 y(suc)m(h)h(sets)h(is)f Fm(O)s Fn(\(1)p Fm(=\017)p Fn(\).)42 b(Hence)31 b(the)g(total)h(n)m(um)m(b)s(er)c(of)j (edges)g(within)e(all)j(sets)e Fm(V)2921 1311 y Fk(i)2980 1297 y Fn(is)g Fm(O)s Fn(\()p Fm(\017m)p Fn(\),)h(and)f(their)150 1410 y(con)m(tribution)i(to)g(the)g(\014nal)e(cut)i(is)f(negligible.)46 b(F)-8 b(or)32 b(the)f(curren)m(t)g(sk)m(etc)m(h)i(of)f(pro)s(of,)f(w)m (e)h(shall)f(assume)150 1523 y(that)39 b(there)f(are)h(no)f(edges)h(at) g(all)g(within)e(the)i(sets)f Fm(V)2086 1537 y Fk(i)2114 1523 y Fn(.)65 b(This)37 b(simpli\014es)g(the)h(presen)m(tation)h(of)g (the)150 1636 y(pro)s(of.)h(A)29 b(more)f(detailed)i(pro)s(of)e(that)i (tak)m(es)g(in)m(to)g(accoun)m(t)h(the)e(e\013ect)h(of)g(these)f(edges) h(can)f(b)s(e)f(found)150 1748 y(in)i([9)q(].)291 1861 y(W)-8 b(e)34 b(call)g(a)e(v)m(ertex)i Fm(v)e Fl(2)c Fm(V)1192 1875 y Fk(i)1253 1861 y Ff(misplac)-5 b(e)g(d)35 b Fn(if)d(it)h(is)g(in)f Fm(A)2110 1875 y Fk(i)p Fi(\000)p Fj(1)2244 1797 y Fe(T)2328 1861 y Fm(B)2397 1875 y Fk(i)2425 1861 y Fn(,)h(or)g(in)f Fm(B)2774 1875 y Fk(i)p Fi(\000)p Fj(1)2907 1797 y Fe(T)2992 1861 y Fm(A)3060 1875 y Fk(i)3088 1861 y Fn(.)47 b(Note)34 b(that)f(only)150 1974 y(misplaced)39 b(v)m(ertices)i(con)m(tribute)f(to)g(the)g(di\013erence)g(b)s(et)m(w)m (een)g(\()p Fm(A)2528 1988 y Fk(i)2556 1974 y Fm(;)15 b(B)2665 1988 y Fk(i)2694 1974 y Fn(\))40 b(and)f(\()p Fm(A)3058 1988 y Fk(i)p Fi(\000)p Fj(1)3176 1974 y Fm(;)15 b(B)3285 1988 y Fk(i)p Fi(\000)p Fj(1)3404 1974 y Fn(\).)68 b(F)-8 b(or)41 b(a)150 2087 y(v)m(ertex)34 b Fm(v)e Fl(2)d Fm(V)647 2101 y Fk(i)675 2087 y Fn(,)k(let)h Fm(d)914 2101 y Fk(A)971 2087 y Fn(\()p Fm(v)s Fn(\))g(denote)f(the)g(n)m(um)m (b)s(er)d(of)j(neigh)m(b)s(ors)f(that)h(it)h(has)e(in)g Fm(A)3060 2101 y Fk(i)p Fi(\000)p Fj(1)3211 2087 y Fn(and)g(let)i Fm(d)3571 2101 y Fk(B)3632 2087 y Fn(\()p Fm(v)s Fn(\))150 2200 y(denote)f(the)f(n)m(um)m(b)s(er)f(of)h(neigh)m(b)s(ors)g(that)h (it)g(has)f(in)g Fm(B)2087 2214 y Fk(i)p Fi(\000)p Fj(1)2205 2200 y Fn(.)47 b(The)32 b(v)m(ertices)i(of)e Fm(V)2955 2214 y Fk(i)3016 2200 y Fn(can)h(b)s(e)e(partitioned)150 2313 y(in)m(to)36 b(three)g(sets:)51 b Ff(str)-5 b(ongly-B)37 b Fn(for)e(whic)m(h)g Fm(d)1696 2327 y Fk(A)1787 2313 y Fm(>)e(d)1938 2327 y Fk(B)2023 2313 y Fn(+)23 b Fm(\017)p Fn(\001,)37 b Ff(str)-5 b(ongly-A)36 b Fn(for)f(whic)m(h)g Fm(d)3191 2327 y Fk(B)3286 2313 y Fm(>)e(d)3437 2327 y Fk(A)3518 2313 y Fn(+)23 b Fm(\017)p Fn(\001,)150 2426 y(and)38 b Ff(neutr)-5 b(al)40 b Fn(for)e(whic)m(h)g Fl(j)p Fm(d)1136 2440 y Fk(A)1219 2426 y Fl(\000)25 b Fm(d)1362 2440 y Fk(B)1423 2426 y Fl(j)39 b(\024)g Fm(\017)p Fn(\001.)64 b(Consider)38 b(the)h(sample)e Fm(S)2719 2440 y Fk(i)2786 2426 y Fn(and)g(a)i(t)m(w)m(o-coloring)j(of)c(it)150 2539 y(consisten)m(t)g(with)f(\()p Fm(A)896 2553 y Fk(i)p Fi(\000)p Fj(1)1015 2539 y Fm(;)15 b(B)1124 2553 y Fk(i)p Fi(\000)p Fj(1)1243 2539 y Fn(\).)60 b(Let)38 b Fm(v)i Fn(b)s(e)c(a)h(v)m(ertex)h(that)g(is)f(strongly-B.)h(The)f(n)m(um)m(b)s (er)d(of)j Fm(B)3601 2553 y Fk(i)p Fi(\000)p Fj(1)3720 2539 y Fn(-)150 2652 y(neigh)m(b)s(ors)k(of)g Fm(v)j Fn(in)d Fm(S)946 2666 y Fk(i)1015 2652 y Fn(is)g(exp)s(ected)h(to)f(b)s (e)g Fm(k)1813 2619 y Fj(3)1852 2652 y Fm(d)1899 2666 y Fk(B)1960 2652 y Fn(\()p Fm(v)s Fn(\))p Fm(=)p Fn(\001.)74 b(The)41 b(n)m(um)m(b)s(er)e(of)i Fm(A)3019 2666 y Fk(i)p Fi(\000)p Fj(1)3138 2652 y Fn(-neigh)m(b)s(ors)f(of)i Fm(v)150 2765 y Fn(in)37 b Fm(S)319 2779 y Fk(i)383 2765 y Fn(is)g(exp)s(ected)h(to)f(b)s(e)g Fm(k)1165 2732 y Fj(3)1204 2765 y Fm(d)1251 2779 y Fk(A)1309 2765 y Fn(\()p Fm(v)s Fn(\))p Fm(=)p Fn(\001)g Fl(\025)f Fm(k)1741 2732 y Fj(3)1781 2765 y Fm(d)1828 2779 y Fk(B)1889 2765 y Fn(\()p Fm(v)s Fn(\))p Fm(=)p Fn(\001)25 b(+)g Fm(k)2298 2732 y Fj(2)2337 2765 y Fn(.)61 b(The)36 b(di\013erence)i(of)f Fm(k)3190 2732 y Fj(2)3266 2765 y Fn(b)s(et)m(w)m(een)h(the)150 2878 y(t)m(w)m(o)j(is)e(statistically)k(signi\014can)m(t)d(b)s(ecause)f Fm(k)1765 2845 y Fj(3)1805 2878 y Fm(d)1852 2892 y Fk(B)1913 2878 y Fn(\()p Fm(v)s Fn(\))p Fm(=)p Fn(\001)j(=)e Fm(O)s Fn(\()p Fm(k)2461 2845 y Fj(3)2500 2878 y Fn(\).)69 b(Hence)40 b Fm(S)2964 2892 y Fk(i)3031 2878 y Fn(will)g(almost)f(surely)150 2990 y(send)d Fm(v)41 b Fn(to)d(side)f Fm(B)825 3004 y Fk(i)853 2990 y Fn(,)i(in)e(whic)m(h)g(case)i(the)e(con)m(tribution)h (of)f(edges)h(connected)g(to)g Fm(v)j Fn(to)d Fl(j)p Fn(\()p Fm(A)3425 3004 y Fk(i)3454 2990 y Fm(;)15 b(B)3563 3004 y Fk(i)3591 2990 y Fn(\))p Fl(j)38 b Fn(is)150 3103 y(at)32 b(least)g(as)g(large)g(as)f(their)g(con)m(tribution)h(to)g Fl(j)p Fn(\()p Fm(A)1900 3117 y Fk(i)p Fi(\000)p Fj(1)2019 3103 y Fm(;)15 b(B)2128 3117 y Fk(i)p Fi(\000)p Fj(1)2247 3103 y Fn(\))p Fl(j)p Fn(.)43 b(The)31 b(fraction)h(of)f(strongly-B)h (v)m(ertices)150 3216 y(that)37 b(are)g(sen)m(t)g(to)g(the)g(side)g Fm(A)1244 3230 y Fk(i)1309 3216 y Fn(is)f(b)s(elo)m(w)h Fm(\017)f Fn(with)h(high)f(probabilit)m(y)h(\(o)m(v)m(er)h(the)f(c)m (hoice)h(of)f Fm(S)3477 3230 y Fk(i)3505 3216 y Fn(\),)i(and)150 3329 y(the)34 b(n)m(um)m(b)s(er)d(of)i(edges)h(that)g(are)g(e\013ected) h(b)m(y)e(these)h(v)m(ertices)h(is)f Fm(O)s Fn(\()p Fm(\017)p Fl(j)p Fm(V)2699 3343 y Fj(1)2739 3329 y Fl(j)p Fn(\001\))c(=)g Fm(O)s Fn(\()p Fm(\017)3150 3296 y Fj(2)3190 3329 y Fm(m)p Fn(\).)50 b(A)34 b(similar)150 3442 y(argumen)m(t)22 b(applies)g(to)h(strongly-A)g(v)m(ertices)h(in)e Fm(V)1861 3456 y Fk(i)1889 3442 y Fn(.)38 b(Hence)24 b(w)m(e)e(are)h(left)g(to)g (deal)g(with)f(neutral)g(v)m(ertices.)150 3555 y(If)40 b(a)h(neutral)g(v)m(ertex)g Fm(v)j Fn(is)d(misplaced,)i(then)d(in)g (terms)g(of)h(edges)g(adjacen)m(t)h(to)f Fm(v)s Fn(,)j(the)c(n)m(um)m (b)s(er)f(of)150 3668 y(edges)c(cut)f(in)g(\()p Fm(A)770 3682 y Fk(i)799 3668 y Fm(;)15 b(B)908 3682 y Fk(i)937 3668 y Fn(\))34 b(di\013ers)g(from)f(the)i(n)m(um)m(b)s(er)d(of)i (edges)h(cut)g(in)f(\()p Fm(A)2724 3682 y Fk(i)p Fi(\000)p Fj(1)2843 3668 y Fm(;)15 b(B)2952 3682 y Fk(i)p Fi(\000)p Fj(1)3070 3668 y Fn(\))35 b(b)m(y)f(at)h(most)f Fm(\017)p Fn(\001,)150 3781 y(whic)m(h)j(totals)i Fm(O)s Fn(\()p Fm(\017)p Fn(\001)p Fl(j)p Fm(V)975 3795 y Fj(1)1014 3781 y Fl(j)p Fn(\))e(=)g Fm(O)s Fn(\()p Fm(\017)1363 3748 y Fj(2)1402 3781 y Fm(m)p Fn(\))h(edges)f(o)m(v)m(er)i(all)f (neutral)f(v)m(ertices)i(in)e Fm(V)2961 3795 y Fk(i)2989 3781 y Fn(.)61 b(Summing)34 b(up)i(the)150 3894 y(con)m(tribution)c(of) g(all)g(misplaced)f(v)m(ertices)i(\(strongly-B,)g(strongly-A)f(and)f (neutral\),)i(w)m(e)e(obtain)h(that)150 4007 y Fl(j)p Fn(\()p Fm(A)278 4021 y Fk(i)307 4007 y Fm(;)15 b(B)416 4021 y Fk(i)445 4007 y Fn(\))p Fl(j)42 b(\025)f(j)p Fn(\()p Fm(A)787 4021 y Fk(i)p Fi(\000)p Fj(1)906 4007 y Fm(;)15 b(B)1015 4021 y Fk(i)p Fi(\000)p Fj(1)1134 4007 y Fn(\))p Fl(j)28 b(\000)e Fm(O)s Fn(\()p Fm(\017)1463 3974 y Fj(2)1503 4007 y Fm(m)p Fn(\),)43 b(as)d(desired.)70 b(This)40 b(completes)g(the)h(sk)m(etc)m(h)g(of)g(pro)s(of)e(for)150 4120 y(Prop)s(osition)30 b(30.)291 4282 y(W)-8 b(e)32 b(w)m(ould)f(no)m(w)f(lik)m(e)i(to)g(pro)m(v)m(e)g(a)f(prop)s(osition)f (similar)g(to)i(Prop)s(osition)f(30)g(for)g Fm(G)3216 4249 y Fi(0)3238 4282 y Fn(.)42 b(F)-8 b(or)32 b(this)e(w)m(e)150 4395 y(need)j(to)h(partition)f(the)g(v)m(ertices)i(of)e Fm(G)1529 4362 y Fi(0)1585 4395 y Fn(in)m(to)h Fm(V)1845 4362 y Fi(0)1825 4418 y Fj(1)1868 4395 y Fm(;)15 b(:)g(:)g(:)i(;)e(V) 2144 4362 y Fi(0)2123 4422 y Fk(k)2167 4395 y Fn(.)49 b(W)-8 b(e)34 b(do)f(it)g(in)g(the)g(unique)f(w)m(a)m(y)i(satisfying) 150 4508 y Fm(V)223 4475 y Fi(0)203 4532 y Fk(i)272 4508 y Fl(\032)25 b Fm(V)421 4522 y Fk(i)474 4508 y Fn(for)g(ev)m(ery)i Fm(i)p Fn(.)39 b(W)-8 b(e)26 b(note)g(that)g(Lemma)e(25)i(can)g(b)s(e)e (used)h(to)h(sho)m(w)f(that)h(with)f(high)g(probabilit)m(y)150 4621 y(o)m(v)m(er)33 b(the)e(c)m(hoice)j(of)d Fm(G)951 4588 y Fi(0)974 4621 y Fn(,)h(the)f Fm(V)1262 4588 y Fi(0)1241 4645 y Fk(i)1316 4621 y Fn(will)h(b)s(e)f(of)g(roughly)h (equal)f(cardinalit)m(y)-8 b(,)34 b(and)d(the)h(n)m(um)m(b)s(er)d(of)i (edges)150 4734 y(in)36 b(subgraphs)e(induced)h(b)m(y)g(eac)m(h)i Fm(V)1457 4701 y Fi(0)1436 4758 y Fk(i)1516 4734 y Fn(is)f Fm(O)s Fn(\()p Fm(\017)1757 4701 y Fj(2)1796 4734 y Fm(m)1876 4701 y Fi(0)1899 4734 y Fn(\).)58 b(Cho)s(osing)35 b(a)i(random)d (sequence)i Fm(S)3283 4701 y Fi(0)3342 4734 y Fn(of)g(v)m(ertices)150 4847 y(in)h Fm(G)335 4814 y Fi(0)395 4847 y Fn(and)g(t)m(w)m (o-coloring)k(it)d(in)f(all)h(2)1491 4814 y Fi(j)p Fk(S)1558 4790 y Fd(0)1580 4814 y Fi(j)1641 4847 y Fn(p)s(ossible)f(w)m(a)m(ys)i (induces)e(2)2594 4814 y Fi(j)p Fk(S)2661 4790 y Fd(0)2683 4814 y Fi(j)2744 4847 y Fn(cuts)g(in)g Fm(G)3128 4814 y Fi(0)3151 4847 y Fn(.)62 b(The)37 b(pro)s(of)g(of)150 4960 y(Prop)s(osition)d(30)g(sho)m(ws)g(that)g(with)g(high)f (probabilit)m(y)-8 b(,)36 b(one)e(of)g(them,)g(sa)m(y)h(\()p Fm(A)2938 4927 y Fi(0)2962 4960 y Fm(;)15 b(B)3076 4927 y Fi(0)3099 4960 y Fn(\),)35 b(is)f Fm(\017)p Fn(-maxim)m(um)1905 5247 y(36)p eop %%Page: 37 37 37 36 bop 150 -58 a Fn(for)30 b Fm(G)361 -91 y Fi(0)384 -58 y Fn(.)291 55 y(No)m(w)39 b(w)m(e)f(reac)m(h)h(an)f(imp)s(ortan)m (t)f(p)s(oin)m(t)h(in)g(the)g(argumen)m(t.)63 b(W)-8 b(e)39 b(w)m(an)m(t)g(the)g(sequences)f Fm(S)43 b Fn(for)38 b Fm(G)150 168 y Fn(and)28 b Fm(S)386 135 y Fi(0)437 168 y Fn(for)g Fm(G)646 135 y Fi(0)697 168 y Fn(to)h(b)s(e)f(the)g (same.)39 b(Then,)28 b(the)h(t)m(w)m(o-coloring)i(of)d Fm(S)34 b Fn(that)28 b(induces)g(the)g Fm(\017)p Fn(-maxim)m(um)e(cut) 150 281 y(\()p Fm(A)253 248 y Fi(0)277 281 y Fm(;)15 b(B)391 248 y Fi(0)414 281 y Fn(\))36 b(in)f Fm(G)668 248 y Fi(0)726 281 y Fn(also)h(induces)f(some)f(cut)i(\()p Fm(A;)15 b(B)5 b Fn(\))36 b(in)f Fm(G)p Fn(.)55 b(The)35 b(imp)s(ortan)m(t)g(p)s(oin)m(t)g(is)g(that)h(\()p Fm(A)3457 248 y Fi(0)3481 281 y Fm(;)15 b(B)3595 248 y Fi(0)3618 281 y Fn(\))36 b(is)150 394 y(precisely)c(the)g(pro)5 b(jection)32 b(on)f Fm(G)1313 361 y Fi(0)1367 394 y Fn(of)h(the)f(cut)h (\()p Fm(A;)15 b(B)5 b Fn(\),)32 b(whic)m(h)g(is)f(the)h(prop)s(ert)m (y)e(needed)h(to)h(complete)150 506 y(the)f(pro)s(of)e(of)i(Lemma)d (28.)291 619 y(Wh)m(y)36 b(can)h(w)m(e)f(tak)m(e)h Fm(S)1091 586 y Fi(0)1150 619 y Fn(to)g(b)s(e)e(the)h(same)g(as)g Fm(S)5 b Fn(?)57 b(Note)37 b(that)f Fm(n)2552 586 y Fi(0)2575 619 y Fm(=)p Fn(\001)2696 586 y Fi(0)2754 619 y Fn(=)e(\002\()p Fm(pn=p)p Fn(\001\))g(=)h(\002\()p Fm(n=)p Fn(\001\).)150 732 y(Hence)c(the)g(in)m(tended)g(lengths)g(of)f Fm(S)36 b Fn(and)30 b Fm(S)1696 699 y Fi(0)1750 732 y Fn(are)h(similar.)40 b(The)31 b(main)e(problem)g(is)i(that)g(if)f Fm(S)36 b Fn(is)31 b(\014xed)150 845 y(prior)26 b(to)g(the)h(c)m(hoice)h(of)e Fm(G)1065 812 y Fi(0)1088 845 y Fn(,)h(then)f(the)g(v)m(ertices)i(of)e Fm(S)32 b Fn(need)25 b(not)i(b)s(e)e(con)m(tained)j(at)f(all)g(in)f Fm(G)3302 812 y Fi(0)3324 845 y Fn(.)40 b(Ho)m(w)m(ev)m(er,)150 958 y(the)28 b(pro)s(of)f(that)h(\(with)f(high)g(probabilit)m(y\))i Fm(S)j Fn(is)c(go)s(o)s(d)f(for)h Fm(G)2274 925 y Fi(0)2324 958 y Fn(w)m(orks)f(an)m(yw)m(a)m(y)-8 b(.)42 b(This)27 b(can)h(b)s(e)e(formally)150 1071 y(sho)m(wn)i(b)m(y)h(considering)g(a) g(new)g(graph)f Fm(G)1611 1038 y Fi(00)1682 1071 y Fn(whose)h(v)m (ertex)h(set)f Fm(V)2436 1038 y Fi(00)2508 1071 y Fn(is)g(the)g(union)f (o)m(v)m(er)i Fm(V)3270 1038 y Fi(0)3322 1071 y Fn(and)f Fm(S)5 b Fn(.)40 b(W)-8 b(e)150 1184 y(partition)29 b Fm(G)599 1151 y Fi(00)670 1184 y Fn(in)m(to)h Fm(k)h Fn(parts)e Fm(V)1236 1151 y Fi(00)1216 1207 y Fj(1)1279 1184 y Fm(;)15 b(:)g(:)g(:)h(;)f(V)1554 1151 y Fi(00)1533 1210 y Fk(k)1625 1184 y Fn(where)28 b Fm(V)1959 1151 y Fi(00)1939 1208 y Fk(i)2030 1184 y Fn(is)h(the)g(pro)5 b(jection)30 b(of)e Fm(V)2860 1198 y Fk(i)2917 1184 y Fn(on)h Fm(G)3114 1151 y Fi(00)3156 1184 y Fn(.)40 b(Note)30 b(that)f Fm(V)3708 1151 y Fi(00)3687 1208 y Fk(i)150 1297 y Fn(is)i(iden)m(tical)j(to)e Fm(V)795 1264 y Fi(0)775 1320 y Fj(1)850 1297 y Fn(except)g(for)f(some)g(v)m(ertices)i(of)e Fm(S)37 b Fn(that)32 b(en)m(ter)g Fm(V)2534 1264 y Fi(00)2514 1321 y Fk(i)2576 1297 y Fn(.)44 b(F)-8 b(or)32 b Fm(G)2881 1264 y Fi(00)2954 1297 y Fn(w)m(e)g(can)g(tak)m(e)h Fm(S)3516 1264 y Fi(0)3566 1297 y Fn(=)27 b Fm(S)5 b Fn(.)150 1410 y(The)28 b(sequence)h Fm(S)771 1377 y Fi(0)823 1410 y Fn(is)g(indeed)f(a)i(random)d(sequence)i(of)g(v)m(ertices)h(in)f Fm(G)2587 1377 y Fi(00)2657 1410 y Fn(\(b)s(ecause)g(b)s(oth)f Fm(G)3312 1377 y Fi(0)3364 1410 y Fn(and)g Fm(S)34 b Fn(are)150 1523 y(random\).)57 b(Hence)38 b(the)e(pro)s(of)g(of)g(Prop) s(osition)h(30)g(do)s(es)f(go)h(through,)h(and)d(an)i Fm(\017)p Fn(-maxim)m(um)c(cut)k(of)150 1636 y Fm(G)222 1603 y Fi(00)299 1636 y Fn(is)f(obtained)g(as)g(a)g(pro)5 b(jection)36 b(of)g(a)g(cut)g(of)f Fm(G)p Fn(.)56 b(No)m(w)37 b(it)f(remains)e(to)i(observ)m(e)h(that)f Fm(G)3396 1603 y Fi(0)3454 1636 y Fn(and)f Fm(G)3708 1603 y Fi(00)150 1748 y Fn(di\013er)c(in)h(at)g(most)f Fl(j)p Fm(S)5 b Fl(j)32 b Fn(v)m(ertices,)i(and)d(this)g(is)h(a)g(lo)m(w)g(order)f (term)g(compared)f(to)j Fm(\017n)3098 1715 y Fi(0)3152 1748 y Fn(\(assuming)d(that)150 1861 y Fm(k)200 1828 y Fj(5)265 1861 y Fl(\034)25 b Fn(log)r(\()p Fm(n=)p Fn(\001\),)30 b(whic)m(h)d(is)h(indeed)g(true)g(b)s(ecause)f(in)h(our)g (in)m(tended)f(application)j(w)m(e)e(ma)m(y)g(tak)m(e)h Fm(n=)p Fn(\001)150 1974 y(to)37 b(b)s(e)g(arbitrarily)f(large\).)62 b(Prop)s(osition)36 b(26)i(implies)e(that)h Fm(G)2367 1941 y Fi(00)2446 1974 y Fn(is)g(also)g(nearly)g(regular.)60 b(It)37 b(follo)m(ws)150 2087 y(that)31 b(the)g(maxim)m(um)c(cuts)j(in) g Fm(G)1298 2054 y Fi(0)1351 2087 y Fn(and)g Fm(G)1600 2054 y Fi(00)1672 2087 y Fn(di\013er)g(b)m(y)h Fm(O)s Fn(\()p Fm(\017m)2258 2054 y Fi(0)2281 2087 y Fn(\))g(edges,)g(whic)m (h)f(can)h(b)s(e)e(neglected.)118 b Fa(2)1905 5247 y Fn(37)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF