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3729 y(w)n(eigh)n(t)j (at)g(least)g(half)g(the)h(optim)n(um:)g(tak)n(e)f Fj(any)g Ft(linear)g(ordering)f(of)h(the)h(v)n(ertices;)e(par-)523 3828 y(tition)37 b(the)g(edges)f(in)n(to)h(t)n(w)n(o)f(sets,)g(those)g (going)g(forw)n(ard)f(in)i(the)g(ordering)e(and)h(those)523 3928 y(going)27 b(bac)n(kw)n(ard.)e(Both)j(sets)f(are)g(acyclic;)g(one) g(of)h(these)f(sets)h(has)f(w)n(eigh)n(t)g(at)h(least)f(half)523 4028 y(the)j(total)f(w)n(eigh)n(t)g(of)g(all)h(the)f(edges)g(in)h(the)g (graph)e(\(and)i(hence)f(at)h(least)f(half)g(the)h(opti-)523 4127 y(m)n(um\).)24 b(This)f(simple)g(algorithm)f(giv)n(es)g(the)i(b)r (est-kno)n(wn)f(p)r(olynomial-time)f(computable)523 4227 y(appro)n(ximation)k(factor)h(for)g(the)h(problem)f(\(namely)2238 4194 y Fs(1)p 2238 4208 34 4 v 2238 4256 a(2)2281 4227 y Ft(\).)648 4327 y(In)19 b(this)h(pap)r(er,)f(w)n(e)h(study)g(the)g (qualit)n(y)f(of)g(p)r(olyhedral)g(relaxations)f(for)h(this)h (optimiza-)523 4426 y(tion)h(problem)g([4,)13 b(5].)21 b(The)g(qualit)n(y)g(of)g(a)f(relaxation)g(can)g(b)r(e)i(measured)e(b)n (y)h(the)g Fj(inte)l(gr)l(ality)523 4526 y(gap)p Ft(,)26 b(the)g(maxim)n(um)e(p)r(ossible)h(ratio)f(b)r(et)n(w)n(een)h(the)g (linear)f(programming)f(optim)n(um)i(and)523 4625 y(the)i(true)f(in)n (tegral)f(optim)n(um.)i(A)g(w)n(ell-kno)n(wn)e(linear)h(programming)e (relaxation)h(for)h(the)523 4725 y(problem)i(is)h(based)f(on)g(the)h (simple)g(idea)f(of)g(requiring)g(that)g(from)h(ev)n(ery)e(directed)h (cycle)523 4825 y Fi(C)6 b Ft(,)21 b(a)f(solution)g(con)n(tains)g(at)g (most)h Fh(j)p Fi(C)6 b Fh(j)t(\000)t Ft(1)20 b(edges.)g(The)h(corresp) r(onding)e(linear)g(constrain)n(ts)523 4924 y(are)25 b(exp)r(onen)n(tial)g(in)h(n)n(um)n(b)r(er)f(\(one)h(for)f(eac)n(h)g (cycle\),)g(but)i(can)e(b)r(e)h(solv)n(ed)f(in)h(p)r(olynomial)p eop %%Page: 2 2 2 1 bop 523 232 a Fq(2)237 b(Alan)n(tha)26 b(Newman)f(and)g(San)n(tosh) g(V)-6 b(empala)523 448 y Ft(time)30 b(via)f(an)h(e\016cien)n(t)f (separation)f(oracle.)h(This)g(and)h(another)e(w)n(ell-kno)n(wn)h (relaxation)523 548 y(are)j(describ)r(ed)g(in)h(Section)g(2.)f(Ho)n(w)h (go)r(o)r(d)f(are)f(the)j(relaxations?)d(The)h(in)n(tegralit)n(y)g(gap) 523 648 y(for)22 b(b)r(oth)i(of)e(these)h(standard)f(relaxations)f (turns)i(out)g(to)f(b)r(e)i(at)e(least)h(2)9 b Fh(\000)g Fi(\017)22 b Ft(for)g(an)n(y)g Fi(\017)h(>)f Ft(0.)523 747 y(Th)n(us,)i(the)h(estimate)f(they)h(pro)n(vide)e(on)h(the)g(optim) n(um)h(is)f(no)g(b)r(etter)h(\(in)g(the)f(w)n(orst-case\))523 847 y(than)k(the)g(trivial)f(upp)r(er)g(b)r(ound)h(of)g(the)g(total)f (edge)g(w)n(eigh)n(t.)648 947 y(A)34 b(natural)g(next)h(step)g(is)f(to) g(strengthen)g(these)h(standard)f(relaxations)e(b)n(y)i(adding)523 1047 y(constrain)n(ts.)19 b(T)-7 b(o)19 b(this)i(end,)f(a)f(promising)g (set)h(of)g(constrain)n(ts)f(are)g(the)h Fi(k)s Fj(-fenc)l(e)g Ft(constrain)n(ts)523 1147 y([4,)14 b(5].)19 b(Although)h(these)g (constrain)n(ts)f(are)g(NP-complete)h(to)f(separate)g(in)h(general)f ([9],)h(they)523 1246 y(can)27 b(b)r(e)h(separated)f(in)h(p)r (olynomial)f(time)h(for)f(an)n(y)g(\014xed)h Fi(k)s Ft(.)g(Another)f (set)h(of)f(constrain)n(ts)523 1346 y(that)34 b(ha)n(v)n(e)f(b)r(een)i (prop)r(osed)e(are)g(the)i Fi(k)s Fj(-M\177)-42 b(obius)35 b Ft(ladder)e(constrain)n(ts)g([4,)13 b(5].)34 b(These)g(are)523 1445 y(kno)n(wn)d(to)g(b)r(e)h(separable)d(in)j(p)r(olynomial)f(time)g ([1,)14 b(12].)30 b(In)i(Section)f(3,)g(w)n(e)g(presen)n(t)g(our)523 1545 y(main)c(result:)g(the)g(in)n(tegralit)n(y)f(gap)g(is)h(2)g(ev)n (en)f(with)i Fi(k)s Ft(-fence)f(constrain)n(ts)e(for)i(an)n(y)f Fi(k)k Ft(and)523 1645 y(with)37 b Fi(k)s Ft(-M\177)-42 b(obius)35 b(ladder)h(constrain)n(ts)e(for)i Fi(k)k Fh(\024)d Ft(7;)f(when)g(augmen)n(ted)g(with)h Fi(k)s Ft(-M\177)-42 b(obius)523 1744 y(ladder)37 b(constrain)n(ts)g(for)h(arbitrary)e Fi(k)s Ft(,)i(the)g(gap)f(is)h(at)g(least)2571 1712 y Fs(33)p 2571 1726 67 4 v 2571 1773 a(17)2688 1744 y Fh(\031)i Ft(1)p Fi(:)p Ft(94.)c(Our)i(pro)r(ofs)523 1844 y(of)e(the)h(in)n (tegralit)n(y)e(gap)g(start)h(with)h(a)f(probabilistic)f(construction)h (whic)n(h)g(is)g(molded)523 1944 y(to)g(ha)n(v)n(e)e(the)i(desired)f (structure)g(\(th)n(us)h(w)n(e)f(demonstrate)g(the)h(existence)f(of)h (extremal)523 2043 y(graphs)26 b(without)i(explicitly)g(describing)f (them\).)648 2144 y(Finally)-7 b(,)34 b(w)n(e)g(establish)g(a)f (concrete)h(lo)n(w)n(er)f(b)r(ound)h(on)g(appro)n(ximabilit)n(y:)f(it)h (is)g(NP-)523 2243 y(hard)k(to)h(appro)n(ximate)e(the)i(optim)n(um)g (to)g(within)g(a)f(factor)g(b)r(etter)h(than)3034 2211 y Fs(65)p 3034 2225 V 3034 2272 a(66)3110 2243 y Ft(,)g(i.e.)g(no)523 2343 y(p)r(olynomial-time)d(solv)-5 b(able)36 b(relaxation)f(can)i(ha)n (v)n(e)e(an)h(in)n(tegralit)n(y)g(gap)g(less)g(than)3306 2310 y Fs(66)p 3306 2324 V 3306 2372 a(65)3382 2343 y Ft(.)523 2443 y(The)22 b(reduction,)g(describ)r(ed)f(in)h(Section)g(4,) g(is)g(from)f(the)h(problem)g(of)g(\014nding)g(a)f(maxim)n(um)523 2542 y(satis\014able)27 b(subset)g(of)h(a)f(giv)n(en)g(set)h(of)f (linear)g(equations)g(mo)r(dulo)g(2.)523 2812 y Fk(2)112 b(Standard)39 b(LP)e(Relaxations)523 3016 y Ft(In)26 b(this)g(section,)g(w)n(e)g(describ)r(e)f(t)n(w)n(o)h(standard)e (linear)i(programming)e(relaxations,)g(pro)n(v)n(e)523 3116 y(that)h(they)g(ha)n(v)n(e)f(the)i(same)e(optimal)h(v)-5 b(alue)25 b(for)f(an)n(y)g(graph,)g(and)h(sho)n(w)f(that)h(b)r(oth)g (relax-)523 3216 y(ations)32 b(can)h(b)r(e)g(arbitrarily)e(close)h(to)g (t)n(wice)h(the)g(v)-5 b(alue)33 b(of)g(the)g(optim)n(um)g(in)g(the)g (w)n(orst)523 3315 y(case.)523 3569 y Fg(2.1)95 b(LP)863 3581 y Ff(1)523 3739 y Ft(The)24 b(maxim)n(um)f(acyclic)g(subgraph)g (problem)g(can)g(b)r(e)h(view)n(ed)f(as)g(maximizing)g(the)h(n)n(um-) 523 3839 y(b)r(er)19 b(of)f(edges)g(sub)5 b(ject)19 b(to)f(a)g (constrain)n(t)f(for)h(ev)n(ery)g(cycle.)g(Gr\177)-42 b(otsc)n(hel,)18 b(J)r(\177)-44 b(unger,)17 b(and)h(Reinelt)523 3939 y(refer)26 b(to)g(these)h(constrain)n(ts)e(as)g Fj(dicycle)31 b(ine)l(qualities)d Ft([4].)e(W)-7 b(e)27 b(will)f(call)g(them)h Fj(cycle)j(c)l(on-)523 4038 y(str)l(aints)p Ft(.)25 b(The)h(constrain)n(ts)e(sp)r(ecify)i(that)g(the)g(sum)g(of)g (the)g(edge)f(v)-5 b(ariables)24 b(on)h(an)n(y)g(cycle)523 4138 y Fi(C)34 b Ft(is)27 b(at)h(most)f Fh(j)p Fi(C)6 b Fh(j)19 b(\000)f Ft(1.)523 4325 y(maximize)1016 4263 y Fe(P)1104 4350 y Fs(\()p Fd(i;j)s Fs(\))p Fc(2)p Fd(E)1392 4325 y Fi(w)1451 4337 y Fd(ij)1510 4325 y Fi(x)1557 4337 y Fd(ij)523 4426 y Ft(sub)5 b(ject)28 b(to:)1023 4464 y Fe(P)1111 4551 y Fd(ij)s Fc(2)p Fd(C)1327 4526 y Fi(x)1374 4538 y Fd(ij)1456 4526 y Fh(\024)22 b(j)p Fi(C)6 b Fh(j)19 b(\000)f Ft(1)83 b Fh(8)p Fi(C)1023 4638 y(x)1070 4650 y Fd(ij)1152 4638 y Fh(2)24 b(f)p Ft(0)p Fi(;)14 b Ft(1)p Fh(g)81 b(8)p Fi(ij)26 b Fh(2)e Fi(E)648 4825 y Ft(The)g(solutions)f (to)h(this)g(in)n(teger)f(program)f(are)h(acyclic)g(subgraphs.)g(It)h (is)g(NP-hard)f(to)523 4924 y(solv)n(e)28 b(this)h(in)n(teger)f (program.)f(Ho)n(w)n(ev)n(er,)g(w)n(e)h(can)h(relax)e(the)j(requiremen) n(t)e(that)h(the)g Fi(x)3346 4936 y Fd(ij)p eop %%Page: 3 3 3 2 bop 1487 232 a Fq(On)25 b(Relaxations)h(for)h(the)e(Linear)h (Ordering)g(Problem)237 b(3)523 448 y Ft(are)29 b(in)h Fh(f)p Ft(0)p Fi(;)14 b Ft(1)p Fh(g)28 b Ft(and)i(replace)f(it)h(with)h (the)f(requiremen)n(t)f(that)h(0)d Fh(\024)f Fi(x)2719 460 y Fd(ij)2805 448 y Fh(\024)h Ft(1.)j(W)-7 b(e)30 b(refer)f(to)523 548 y(this)22 b(linear)f(programming)f(relaxation)g (as)h(LP)2000 560 y Fs(1)2036 548 y Ft(.)h(W)-7 b(e)22 b(can)g(solv)n(e)e(LP)2671 560 y Fs(1)2729 548 y Ft(in)i(p)r(olynomial) f(time)523 648 y(using)32 b(the)h(Ellipsoid)e(Algorithm)h([6])g(via)g (the)h(follo)n(wing)e(p)r(olynomial-time)h(separation)523 747 y(oracle.)h(Giv)n(en)h(an)g(assignmen)n(t)f(for)g(the)i(v)-5 b(ariables)33 b Fi(x)2277 759 y Fd(ij)2336 747 y Ft(,)h(w)n(e)g (consider)f(the)h(graph)f(with)523 847 y(eac)n(h)24 b(edge)h(\()p Fi(i;)14 b(j)5 b Ft(\))25 b(assigned)f(a)h(w)n(eigh)n(t)f(of)h(1)14 b Fh(\000)g Fi(x)2015 859 y Fd(ij)2072 847 y Ft(.)26 b(In)f(this)g(graph,)f(w)n(e)h(\014nd)h(the)f(minim)n(um)523 946 y(w)n(eigh)n(t)c(cycle.)h(If)g(there)g(is)g(an)n(y)f(cycle)g(with)i (w)n(eigh)n(t)e(less)h(than)g(1,)f(then)i(the)f(corresp)r(onding)523 1046 y(cycle)e(of)g(length)g Fi(C)26 b Ft(actually)20 b(has)f(w)n(eigh)n(t)h(more)f(than)h Fi(C)9 b Fh(\000)s Ft(1,)21 b(whic)n(h)f(highligh)n(ts)f(a)h(violated)523 1146 y(constrain)n(t.)523 1391 y Fg(2.2)95 b(LP)863 1403 y Ff(2)523 1553 y Ft(Another)18 b(in)n(teger)g(program)e(is)i(based)g (on)g(the)h(linear)f(ordering)e(problem.)i(It)h(has)f(a)g(v)-5 b(ariable)523 1653 y(for)25 b(ev)n(ery)f(pair)g(of)h(v)n(ertices)f Fi(i;)14 b(j)28 b Fh(2)23 b Fi(V)c Ft(.)26 b(In)f(this)g(program,)f (there)h(are)f(only)h(constrain)n(ts)f(for)523 1753 y(2-)33 b(and)g(3-cycles.)f(This)i(set)f(of)h(constrain)n(ts)e(is)h(discussed)g (b)n(y)g(Gr\177)-42 b(otsc)n(hel,)33 b(J)r(\177)-44 b(unger,)32 b(and)523 1852 y(Reinelt)c([5].)523 2030 y(maximize)1016 1968 y Fe(P)1104 2055 y Fd(ij)1176 2030 y Fi(w)1235 2042 y Fd(ij)1294 2030 y Fi(x)1341 2042 y Fd(ij)523 2130 y Ft(sub)5 b(ject)28 b(to:)1070 2230 y Fi(x)1117 2242 y Fd(ij)1195 2230 y Ft(+)18 b Fi(x)1325 2242 y Fd(j)s(i)1407 2230 y Ft(=)k(1)83 b Fh(8)p Fi(i;)14 b(j)27 b Fh(2)c Fi(V)1070 2341 y(x)1117 2353 y Fd(ij)1195 2341 y Ft(+)18 b Fi(x)1325 2353 y Fd(j)s(k)1415 2341 y Ft(+)g Fi(x)1545 2353 y Fd(k)q(i)1633 2341 y Fh(\024)23 b Ft(2)82 b Fh(8)p Fi(i;)14 b(j;)g(k)25 b Fh(2)f Fi(V)1070 2453 y(x)1117 2465 y Fd(ij)1199 2453 y Fh(2)g(f)p Ft(0)p Fi(;)14 b Ft(1)p Fh(g)81 b(8)p Fi(i;)14 b(j)26 b Fh(2)e Fi(V)648 2631 y Ft(Solutions)f(for)g(this)h(in)n(teger)f(program)f(corresp)r (ond)g(to)i(linear)f(orderings.)f(Again,)h(it)h(is)523 2730 y(NP-hard)g(to)g(solv)n(e)g(this)h(in)n(teger)f(program.)f(W)-7 b(e)25 b(refer)f(to)h(the)g(corresp)r(onding)e(relaxation)523 2830 y(as)g(LP)730 2842 y Fs(2)766 2830 y Ft(.)h(Although)f(LP)1284 2842 y Fs(2)1344 2830 y Ft(only)g(con)n(tains)g(constrain)n(ts)e(for)i (2-)g(and)g(3-cycles,)f(w)n(e)h(can)g(sho)n(w)523 2929 y(that)28 b(a)f(v)-5 b(alid)28 b(solution)f(for)g(LP)1523 2941 y Fs(2)1587 2929 y Ft(do)r(es)g(not)h(violate)f Fj(any)h Ft(cycle)f(constrain)n(ts.)523 3099 y Fg(Lemma)j(1.)40 b Fj(A)30 b(solution)g(for)g(LP)1623 3111 y Fs(2)1691 3099 y Fj(do)l(es)g(not)g(violate)h(any)f(cycle)h(c)l(onstr)l(aints.) 523 3269 y(Pr)l(o)l(of.)43 b Ft(W)-7 b(e)27 b(will)g(pro)n(v)n(e)e(b)n (y)h(induction)h(on)f Fi(k)j Ft(that)e(a)f(solution)g(for)g(LP)2767 3281 y Fs(2)2830 3269 y Ft(do)r(es)g(not)h(violate)523 3369 y(an)n(y)e Fi(k)s Ft(-cycle)h(constrain)n(ts.)e(Clearly)-7 b(,)25 b(a)h(v)-5 b(alid)26 b(solution)g(for)f(LP)2517 3381 y Fs(2)2580 3369 y Ft(do)r(es)h(not)g(violate)f(an)n(y)h(2-)523 3468 y(or)f(3-cycle)h(constrain)n(ts.)e(Assume)j(all)f Fi(k)s Ft(-cycle)f(constrain)n(ts)g(are)g(satis\014ed.)h(Then)h(w)n(e)f (will)523 3568 y(sho)n(w)e(that)h(all)g(\()p Fi(k)16 b Ft(+)d(1\)-cycle)23 b(constrain)n(ts)h(are)g(satis\014ed.)g(Consider) g(a)g(cycle)h Fi(C)31 b Ft(of)25 b(length)523 3667 y Fi(k)11 b Ft(+)e(1.)22 b(Cho)r(ose)g(t)n(w)n(o)g(non-adjacen)n(t)g(v)n (ertices)g Fi(i;)14 b(j)27 b Ft(in)c Fi(C)6 b Ft(.)23 b(No)n(w)f(consider)g(the)h(follo)n(wing)f(t)n(w)n(o)523 3767 y(edge-disjoin)n(t)k(cycles:)g Fi(C)1326 3779 y Fs(1)1391 3767 y Ft(is)g(comp)r(osed)g(of)h(edge)f(\()p Fi(i;)14 b(j)5 b Ft(\))27 b(and)g(the)g(path)g(from)f Fi(j)32 b Ft(to)26 b Fi(i)h Ft(in)g Fi(C)523 3867 y Ft(and)i Fi(C)745 3879 y Fs(2)812 3867 y Ft(is)g(comp)r(osed)g(of)g(edge)g(\()p Fi(j;)14 b(i)p Ft(\))30 b(and)f(the)h(path)g(from)f Fi(i)g Ft(to)g Fi(j)34 b Ft(in)c Fi(C)6 b Ft(.)30 b(By)f(induction,)523 3966 y(w)n(e)j(ha)n(v)n(e)846 3904 y Fe(P)933 3991 y Fd(e)p Fc(2)p Fd(C)1057 3999 y Fb(1)1108 3966 y Fi(x)1155 3978 y Fd(e)1212 3966 y Ft(+)1298 3904 y Fe(P)1386 3991 y Fd(e)p Fc(2)p Fd(C)1510 3999 y Fb(2)1560 3966 y Fi(x)1607 3978 y Fd(e)1674 3966 y Fh(\024)e(j)p Fi(C)1851 3978 y Fs(1)1889 3966 y Fh(j)21 b(\000)g Ft(1)g(+)g Fh(j)p Fi(C)2250 3978 y Fs(2)2288 3966 y Fh(j)g(\000)g Ft(1)31 b Fh(\024)f(j)p Fi(C)6 b Fh(j)32 b Ft(and)g Fi(x)2942 3978 y Fd(ij)3023 3966 y Ft(+)21 b Fi(x)3156 3978 y Fd(j)s(i)3245 3966 y Ft(=)30 b(1.)523 4066 y(Th)n(us,)756 4004 y Fe(P)844 4091 y Fd(e)p Fc(2)p Fd(C)990 4066 y Fi(x)1037 4078 y Fd(e)1096 4066 y Ft(=)1184 4004 y Fe(P)1271 4091 y Fd(e)p Fc(2)p Fd(C)1395 4099 y Fb(1)1446 4066 y Fi(x)1493 4078 y Fd(e)1547 4066 y Ft(+)1630 4004 y Fe(P)1718 4091 y Fd(e)p Fc(2)p Fd(C)1842 4099 y Fb(2)1892 4066 y Fi(x)1939 4078 y Fd(e)1994 4066 y Fh(\000)18 b Fi(x)2124 4078 y Fd(ij)2201 4066 y Fh(\000)g Fi(x)2331 4078 y Fd(j)s(i)2413 4066 y Fh(\024)23 b(j)p Fi(C)6 b Fh(j)19 b(\000)f Ft(1.)571 b Fh(u)-55 b(t)648 4236 y Ft(A)27 b(maxim)n(um)g(acyclic)g(subgraph)f (has)h(the)h(same)e(w)n(eigh)n(t)h(as)g(a)g(maxim)n(um)g(linear)f(or-) 523 4335 y(dering,)e(i.e.)g(the)h(optimal)f(in)n(tegral)f(solutions)h (for)g(the)h(t)n(w)n(o)e(in)n(teger)h(programs)e(ab)r(o)n(v)n(e)h(are) 523 4435 y(equal.)j(W)-7 b(e)26 b(no)n(w)g(pro)n(v)n(e)e(that)j(the)g (optimal)f(solutions)f(for)h(the)g(t)n(w)n(o)g(linear)f(programming)523 4535 y(relaxations)g(are)h(equal.)g(F)-7 b(or)26 b(some)g(graph)g Fi(G)d Ft(=)g(\()p Fi(V)5 b(;)14 b(E)5 b Ft(\))27 b(with)h(edge)e(w)n (eigh)n(ts)g Fi(w)f Ft(=)e Fh(f)p Fi(w)3282 4547 y Fd(ij)3340 4535 y Fh(g)p Ft(,)523 4634 y(let)31 b Fi(O)r(P)12 b(T)g Ft(\()p Fi(LP)979 4646 y Fs(1)1016 4634 y Ft(\))32 b(denote)f(an)f (optimal)h(solution)g(for)f(LP)2334 4646 y Fs(1)2402 4634 y Ft(and)h Fh(j)p Fi(O)r(P)12 b(T)g Ft(\()p Fi(LP)2923 4646 y Fs(1)2959 4634 y Ft(\))p Fh(j)32 b Ft(denote)f(its)523 4734 y(ob)5 b(jectiv)n(e)27 b(v)-5 b(alue.)27 b(De\014ne)i Fi(O)r(P)12 b(T)g Ft(\()p Fi(LP)1704 4746 y Fs(2)1740 4734 y Ft(\))28 b(similarly)f(with)h Fi(w)2388 4746 y Fd(ij)2470 4734 y Ft(=)23 b(0)k(for)g(all)g(\()p Fi(i;)14 b(j)5 b Ft(\))33 b Fi(=)-52 b Fh(2)24 b Fi(E)5 b Ft(.)523 4904 y Fg(Theorem)30 b(1.)41 b Fh(j)p Fi(O)r(P)12 b(T)g Ft(\()p Fi(LP)1400 4916 y Fs(1)1437 4904 y Ft(\))p Fh(j)24 b Ft(=)e Fh(j)p Fi(O)r(P)12 b(T)g Ft(\()p Fi(LP)1959 4916 y Fs(2)1996 4904 y Ft(\))p Fh(j)p Fj(.)p eop %%Page: 4 4 4 3 bop 523 232 a Fq(4)237 b(Alan)n(tha)26 b(Newman)f(and)g(San)n(tosh) g(V)-6 b(empala)523 448 y Fj(Pr)l(o)l(of.)43 b Ft(First,)20 b(w)n(e)g(will)g(sho)n(w)f(that)h Fh(j)p Fi(O)r(P)12 b(T)g Ft(\()p Fi(LP)1991 460 y Fs(1)2028 448 y Ft(\))p Fh(j)23 b(\025)g(j)p Fi(O)r(P)12 b(T)g Ft(\()p Fi(LP)2550 460 y Fs(2)2587 448 y Ft(\))p Fh(j)p Ft(,)20 b(i.e.)g(giv)n(en)g(an)f (optimal)523 548 y(solution)25 b(for)f(LP)1068 560 y Fs(2)1105 548 y Ft(,)h(w)n(e)g(can)f(\014nd)i(a)e(solution)h(for)f(LP) 2197 560 y Fs(1)2259 548 y Ft(with)i(the)f(same)g(v)-5 b(alue.)25 b(W)-7 b(e)25 b(simply)523 648 y(let)30 b(the)f(solution)g (for)g(LP)1343 660 y Fs(1)1409 648 y Ft(b)r(e)h(the)f(subset)h(of)f Fh(f)p Fi(x)2110 660 y Fd(ij)2168 648 y Fh(g)g Ft(suc)n(h)g(that)h(\()p Fi(i;)14 b(j)5 b Ft(\))26 b Fh(2)g Fi(E)5 b Ft(.)29 b(By)g(Lemma)523 747 y(1,)34 b(this)h(solution)f(do)r(es)g(not)g(violate)g(an)n(y)f (cycle)h(constrain)n(ts)f(and)h(is)h(therefore)e(a)h(v)-5 b(alid)523 847 y(solution)27 b(for)g(LP)1073 859 y Fs(1)1110 847 y Ft(.)648 946 y(Second,)f(w)n(e)f(will)i(sho)n(w)e(that)i Fh(j)p Fi(O)r(P)12 b(T)g Ft(\()p Fi(LP)1966 958 y Fs(2)2003 946 y Ft(\))p Fh(j)23 b(\025)g(j)p Fi(O)r(P)12 b(T)g Ft(\()p Fi(LP)2525 958 y Fs(1)2562 946 y Ft(\))p Fh(j)p Ft(,)26 b(i.e.)h(giv)n(en)e(an)h(optimal)523 1046 y(solution)31 b(for)h(LP)1082 1058 y Fs(1)1118 1046 y Ft(,)g(w)n(e)g(can)g(construct) f(a)g(solution)h(for)f(LP)2458 1058 y Fs(2)2527 1046 y Ft(with)h(the)g(same)g(ob)5 b(jectiv)n(e)523 1146 y(v)-5 b(alue.)26 b(Assign)g(all)f(edges)h(in)g Fi(E)31 b Ft(v)-5 b(alue)26 b Fi(x)1805 1158 y Fd(ij)1890 1146 y Ft(where)g Fi(x)2176 1158 y Fd(ij)2261 1146 y Ft(is)g(tak)n(en)f(from)h(the)g(giv) n(en)g(solution)523 1245 y(for)33 b(LP)765 1257 y Fs(1)802 1245 y Ft(.)h(Since)g(this)g(is)f(a)g(v)-5 b(alid)34 b(solution)f(for)h(LP)2184 1257 y Fs(1)2220 1245 y Ft(,)g(no)f(cycle)h (constrain)n(ts)e(ha)n(v)n(e)h(b)r(een)523 1345 y(violated)c(th)n(us)h (far.)f(No)n(w)h(consider)f(an)g(arbitrary)f(order)g(for)i(the)g(pairs) f(\()p Fi(j;)14 b(i)p Ft(\))30 b(suc)n(h)g(that)523 1445 y(\()p Fi(i;)14 b(j)5 b Ft(\))35 b Fh(2)f Fi(E)40 b Ft(and)34 b(\()p Fi(j;)14 b(i)p Ft(\))44 b Fi(=)-51 b Fh(2)35 b Fi(E)40 b Ft(and)34 b(assign)f Fi(x)1942 1457 y Fd(j)s(i)2036 1445 y Ft(=)h(1)22 b Fh(\000)h Fi(x)2334 1457 y Fd(ij)2427 1445 y Ft(in)35 b(that)g(order.)e(Let)i(\()p Fi(j;)14 b(i)p Ft(\))35 b(b)r(e)523 1544 y(the)26 b(\014rst)f(edge)g(causing)f (a)i(violated)e(cycle)h(constrain)n(t.)g(Then)g(there)h(is)f(some)g (path)g Fi(p)3254 1556 y Fd(ij)3338 1544 y Ft(of)523 1644 y(length)32 b Fi(`)g Ft(from)g Fi(i)f Ft(to)h Fi(j)38 b Ft(suc)n(h)31 b(that)i(the)f(total)g(v)-5 b(alue)32 b(of)g(the)h(edges)e(in)h Fi(p)2848 1656 y Fd(ij)2939 1644 y Ft(is)g(more)f(than)523 1743 y Fi(`)8 b Fh(\000)g Fi(x)686 1755 y Fd(j)s(i)744 1743 y Ft(.)22 b(Since)h(\()p Fi(i;)14 b(j)5 b Ft(\))23 b Fh(2)g Fi(E)5 b Ft(,)23 b(and)f(the)g (solution)g(for)g(the)h(edges)e(in)i Fi(E)k Ft(is)22 b(optimal,)g(it)h(m)n(ust)g(b)r(e)523 1843 y(the)j(case)f(that)h(there) f(is)h(some)f(path)h Fi(p)1748 1855 y Fd(j)s(i)1832 1843 y Ft(of)g(length)f Fi(`)2211 1813 y Fc(0)2260 1843 y Ft(suc)n(h)g(that)h(the)g(v)-5 b(alue)26 b(of)g(the)g(edges)523 1943 y(in)c Fi(p)656 1955 y Fd(j)s(i)735 1943 y Ft(equals)f(to)g Fi(`)1111 1913 y Fc(0)1140 1943 y Fh(\000)6 b Fi(x)1258 1955 y Fd(ij)1316 1943 y Ft(,)22 b(i.e.)f(a)g(cycle)g(constrain)n(t)f (for)h(some)g(cycle)g(con)n(taining)f(edge)h(\()p Fi(i;)14 b(j)5 b Ft(\))523 2042 y(m)n(ust)33 b(b)r(e)h(tigh)n(t,)g(otherwise)e (w)n(e)h(could)g(increase)f(the)i(v)-5 b(alue)33 b(of)g Fi(x)2649 2054 y Fd(ij)2708 2042 y Ft(.)h(Th)n(us,)f Fi(p)3046 2054 y Fd(ij)3138 2042 y Ft(and)g Fi(p)3347 2054 y Fd(j)s(i)523 2142 y Ft(form)21 b(a)h(cycle)f(of)h(length)g Fi(`)7 b Ft(+)g Fi(`)1462 2112 y Fc(0)1505 2142 y Ft(of)22 b(v)-5 b(alue)21 b(more)g(than)h Fi(`)7 b Ft(+)g Fi(`)2341 2112 y Fc(0)2369 2142 y Fh(\000)g Ft(\()p Fi(x)2520 2154 y Fd(ij)2585 2142 y Ft(+)g Fi(x)2704 2154 y Fd(j)s(i)2762 2142 y Ft(\))24 b(=)e Fi(`)7 b Ft(+)g Fi(`)3054 2112 y Fc(0)3083 2142 y Fh(\000)g Ft(1.)20 b(So)i(it)523 2242 y(is)27 b(a)g(con)n(tradiction)f(that)i(this)f(edge)g(is)g(the)h (\014rst)f(to)g(cause)g(a)g(violated)f(cycle)h(constrain)n(t.)523 2341 y(Also)h(note)g(that)h(all)f(2-cycles)f(in)i Fi(E)34 b Ft(ha)n(v)n(e)27 b(total)h(v)-5 b(alue)28 b(exactly)g(1.)g (Otherwise,)g(for)g(some)523 2441 y Fi(x)570 2453 y Fd(ij)654 2441 y Ft(+)d Fi(x)791 2453 y Fd(j)s(i)890 2441 y Fi(<)39 b Ft(1,)f(w)n(e)f(can)g(\014nd)h(a)g(cycle)f(comp)r(osed)g(of)h(the)g (paths)f Fi(p)2789 2453 y Fd(j)s(i)2885 2441 y Ft(and)h Fi(p)3099 2453 y Fd(ij)3195 2441 y Ft(whic)n(h)523 2540 y(violates)30 b(a)g(cycle)h(constrain)n(t.)e(Simply)i(let)h Fi(p)1974 2552 y Fd(j)s(i)2063 2540 y Ft(b)r(e)f(the)g(path)g(in)g(the) g(cycle)f(with)i Fi(x)3216 2552 y Fd(ij)3306 2540 y Ft(for)523 2640 y(whic)n(h)c(a)f(cycle)g(constrain)n(t)f(is)i(tigh)n(t,)g(and)f (de\014ne)h Fi(p)2176 2652 y Fd(ij)2262 2640 y Ft(similarly)-7 b(.)648 2740 y(Let)816 2719 y(\026)798 2740 y Fi(G)25 b Ft(=)f(\()p Fi(V)5 b(;)1119 2719 y Ft(\026)1099 2740 y Fi(E)h Ft(\))29 b(b)r(e)g(the)g(graph)f(with)h(edge)f(set)2252 2719 y(\026)2233 2740 y Fi(E)i Ft(=)25 b Fh(f)p Ft(\()p Fi(i;)14 b(j)5 b Ft(\))p Fh(g)28 b Ft(suc)n(h)g(that)h(\()p Fi(i;)14 b(j)5 b Ft(\))25 b Fh(2)h Fi(E)523 2839 y Ft(or)g(\()p Fi(j;)14 b(i)p Ft(\))23 b Fh(2)h Fi(E)5 b Ft(.)27 b(By)f(the)h(argumen) n(t)f(ab)r(o)n(v)n(e,)f(all)h(2-cycles)g(in)2441 2818 y(\026)2422 2839 y Fi(G)h Ft(ha)n(v)n(e)e(v)-5 b(alue)27 b(exactly)f(1)g(and)523 2939 y(no)e(cycle)h(constrain)n(ts)e(are)h (violated.)g(No)n(w)g(w)n(e)g(will)h(assign)e(v)-5 b(alues)25 b Fi(x)2723 2951 y Fd(ij)2806 2939 y Ft(to)g(all)f(edges)g(\()p Fi(i;)14 b(j)5 b Ft(\))523 3039 y(suc)n(h)32 b(that)g Fi(i;)14 b(j)35 b Fh(2)30 b Fi(V)51 b Ft(and)32 b(neither)g(\()p Fi(i;)14 b(j)5 b Ft(\))32 b(nor)f(\()p Fi(j;)14 b(i)p Ft(\))33 b(are)e(in)h Fi(E)5 b Ft(.)32 b(W)-7 b(e)32 b(de\014ne)g(the)h Fj(shortest)523 3138 y(p)l(ath)c Ft(b)r(et)n(w)n (een)f Fi(i)g Ft(and)h Fi(j)k Ft(to)28 b(b)r(e)h(the)g(path)f(with)h (the)g(least)f(total)g(v)-5 b(alue,)28 b(where)g Fi(x)3118 3150 y Fd(ij)3205 3138 y Ft(is)h(the)523 3238 y(v)-5 b(alue)27 b(of)g(an)g(edge.)f(Let)h Fi(\013)1358 3250 y Fd(ij)1444 3238 y Ft(b)r(e)h(the)f(length)g(of)g(the)g(shortest)g (directed)g(path)g(from)f Fi(i)h Ft(to)g Fi(j)523 3337 y Ft(in)641 3316 y(\026)623 3337 y Fi(G)p Ft(.)j(De\014ne)h Fi(\013)1054 3349 y Fd(j)s(i)1143 3337 y Ft(similarly)-7 b(.)30 b(F)-7 b(or)30 b(an)n(y)f Fi(i;)14 b(j)35 b Ft(the)c(shortest)e (paths)i(from)f Fi(i)g Ft(to)g Fi(j)35 b Ft(and)30 b(from)523 3437 y Fi(j)j Ft(to)28 b Fi(i)g Ft(form)g(a)f(cycle)h(in)g(the)h (curren)n(t)e(graph.)g(Therefore,)g Fi(\013)2463 3449 y Fd(ij)2540 3437 y Ft(+)18 b Fi(\013)2676 3449 y Fd(j)s(i)2759 3437 y Fh(\025)23 b Ft(1.)28 b(Without)h(loss)523 3537 y(of)i(generalit)n(y)-7 b(,)29 b(assume)h Fi(\013)1366 3549 y Fd(ij)1453 3537 y Fh(\024)f Fi(\013)1600 3549 y Fd(j)s(i)1658 3537 y Ft(.)i(Then)g(let)g Fi(x)2102 3549 y Fd(ij)2190 3537 y Ft(=)d(min)p Fh(f)2473 3504 y Fs(1)p 2473 3518 34 4 v 2473 3565 a(2)2516 3537 y Fi(;)14 b(\013)2606 3549 y Fd(ij)2664 3537 y Fh(g)31 b Ft(and)f Fi(x)2948 3549 y Fd(j)s(i)3036 3537 y Ft(=)e(1)20 b Fh(\000)g Fi(x)3323 3549 y Fd(ij)3382 3537 y Ft(.)523 3636 y(Th)n(us,)29 b(ev)n(ery)f(cycle)i(that)f(includes)h(edge)f(\()p Fi(i;)14 b(j)5 b Ft(\))29 b(or)g(edge)g(\()p Fi(j;)14 b(i)p Ft(\))30 b(will)f(ha)n(v)n(e)g(v)-5 b(alue)29 b(at)g(least)523 3736 y(1,)c(whic)n(h)g(implies)g(that)h(ev)n(ery)e(cycle)h Fi(C)31 b Ft(will)26 b(ha)n(v)n(e)e(v)-5 b(alue)25 b(at)g(most)g Fh(j)p Fi(C)6 b Fh(j)14 b(\000)g Ft(1.)24 b(This)h(implies)523 3836 y(that)f(all)f(3-cycle)g(constrain)n(ts)f(are)h(satis\014ed)g(and) g(all)h(2-cycles)e(ha)n(v)n(e)g(v)-5 b(alue)24 b(exactly)f(1.)83 b Fh(u)-55 b(t)523 4068 y Fg(2.3)95 b(In)m(tegralit)m(y)32 b(Gap)523 4218 y Ft(The)25 b Fj(inte)l(gr)l(ality)k(gap)d Ft(of)f(a)g(linear)g(program)e(is)i(the)h(w)n(orst)e(case)h(ratio)f(b)r (et)n(w)n(een)h(the)h(v)-5 b(alue)523 4318 y(of)33 b(an)f(optimal)g (fractional)g(solution)g(and)g(the)h(v)-5 b(alue)33 b(of)f(an)h (optimal)f(in)n(tegral)g(solution)523 4417 y(o)n(v)n(er)26 b(all)h(w)n(eigh)n(t)g(functions)h Fi(w)e Ft(=)c Fh(f)p Fi(w)1710 4429 y Fd(ij)1769 4417 y Fh(g)p Ft(.)27 b(F)-7 b(ormally)g(,)27 b(the)h(in)n(tegralit)n(y)e(gap)h(is)g(de\014ned)h (as,)1659 4644 y(max)1669 4695 y Fd(w)r(>)p Fs(0)1837 4587 y Fh(j)p Fi(O)r(P)12 b(T)g Ft(\()p Fi(LP)g Ft(\))p Fh(j)p 1837 4624 423 4 v 1844 4700 a(j)p Fi(O)r(P)g(T)g Ft(\()p Fi(I)7 b(P)12 b Ft(\))p Fh(j)523 4825 y Ft(In)29 b(this)g(section,)f(w)n(e)h(sho)n(w)e(that)i(the)g(in)n(tegralit)n(y)f (gap)g(for)g(b)r(oth)h(LP)2740 4837 y Fs(1)2805 4825 y Ft(and)g(LP)3077 4837 y Fs(2)3142 4825 y Ft(is)g(2)18 b Fh(\000)h Fi(\017)523 4924 y Ft(for)j(an)n(y)g Fi(\017)h(>)f Ft(0.)h(As)f(the)h(basis)f(of)h(the)g(construction,)e(w)n(e)i(use)f (the)h(fact)g(that)g(there)f(exists)g(a)p eop %%Page: 5 5 5 4 bop 1487 232 a Fq(On)25 b(Relaxations)h(for)h(the)e(Linear)h (Ordering)g(Problem)237 b(5)523 448 y Ft(class)20 b(of)i(undirected)f (graphs)f(with)i(girth)f Fi(g)j Ft(and)d Fi(\002)r Ft(\()p Fi(n)2212 418 y Fs(1+1)p Fd(=g)2403 448 y Ft(\))g(edges.)g(This)g (result)g(is)h(due)f(to)523 548 y(Erd\177)-42 b(os)26 b(and)h(Sac)n(hs)g([2].)g(Graphs)g(from)g(this)g(class)g(ha)n(v)n(e)f (b)r(een)i(used)f(to)h(pro)n(v)n(e)d(in)n(tegralit)n(y)523 648 y(gaps)g(for)h(the)h(maxim)n(um)f(cut)h(problem)f([11].)g(Based)f (on)h(these)h(graphs,)e(w)n(e)h(de\014ne)g Fi(G)p Ft(\()p Fi(n)p Ft(\))523 747 y(to)32 b(b)r(e)g(a)f(family)g(of)h(graphs)e(with) i(the)g(follo)n(wing)f(prop)r(erties.)g(A)h(graph)e Fi(G)g Fh(2)g Fi(G)p Ft(\()p Fi(n)p Ft(\))j(has)523 847 y Fi(n)c Ft(v)n(ertices,)f(girth)g Fi(g)g Ft(=)d Fi(\002)r Ft(\()1445 810 y Fs(log)12 b Fd(n)p 1396 828 236 4 v 1396 875 a Fs(log)h(log)f Fd(n)1642 847 y Ft(\),)29 b(and)g Fi(n)1939 817 y Fs(1+1)p Fd(=g)2157 847 y Ft(edges.)g(Then)g(w)n(e)f(ha)n(v)n(e)g (the)h(follo)n(wing)523 946 y(lemma.)523 1090 y Fg(Lemma)h(2.)40 b Fj(F)-6 b(or)42 b(any)f Fi(\017)i(>)h Ft(0)p Fj(,)d(ther)l(e)g (exists)g(an)g Fi(n)i Fh(\025)g Fi(f)9 b Ft(\()p Fi(\017)p Ft(\))41 b Fj(such)g(that)g(at)g(le)l(ast)h(one)523 1190 y(dir)l(e)l(cte)l(d)36 b(orientation)g(of)h Fi(G)c Ft(=)g(\()p Fi(V)5 b(;)14 b(E)5 b Ft(\))34 b Fh(2)g Fi(G)p Ft(\()p Fi(n)p Ft(\))i Fj(has)g(the)g(fol)t(lowing)h(pr)l(op)l(erty:)g(for)g (any)523 1289 y(or)l(dering)31 b(of)f(the)g(vertic)l(es,)h(the)f(numb)l (er)f(of)i(forwar)l(d)g(e)l(dges)f(is)g(at)g(most)g Ft(\(1)18 b(+)g Fi(\017)p Ft(\))p Fh(j)p Fi(E)5 b Fh(j)p Fi(=)p Ft(2)p Fj(.)648 1433 y Ft(A)36 b(pro)r(of)g(of)g(Lemma)g(2)g(can)g(b)r (e)g(found)h(in)g([10].)e(W)-7 b(e)37 b(no)n(w)e(de\014ne)i Fi(G)p Ft(\()p Fi(\017)p Ft(\))g(to)f(b)r(e)h(the)523 1532 y(family)42 b(of)f Fj(dir)l(e)l(cte)l(d)h Ft(graphs)e(on)h Fi(n)46 b Fh(\025)g Fi(f)9 b Ft(\()p Fi(\017)p Ft(\))41 b(v)n(ertices)f(whose)h(underlying)g(undirected)523 1632 y(graphs)32 b(b)r(elong)i(to)f Fi(G)p Ft(\()p Fi(n)p Ft(\))i(and)e(whic)n(h)h(ha)n(v)n(e)e(maxim)n(um)i(acyclic)f(subgraphs) f(of)i(size)f(at)523 1732 y(most)d(\(1)20 b(+)g Fi(\017)p Ft(\))p Fh(j)p Fi(E)5 b Fh(j)p Fi(=)p Ft(2.)29 b(All)i(edges)e(in)i Fi(G)d Fh(2)f Fi(G)p Ft(\()p Fi(\017)p Ft(\))k(ha)n(v)n(e)e(w)n(eigh)n (t)h Fi(w)2573 1744 y Fd(ij)2659 1732 y Ft(=)d(1.)j(W)-7 b(e)30 b(use)g Fi(G)p Ft(\()p Fi(\017)p Ft(\))h(to)523 1831 y(pro)n(v)n(e)26 b(the)i(follo)n(wing)e(theorem.)523 1975 y Fg(Theorem)k(2.)41 b Fj(The)31 b(inte)l(gr)l(ality)f(gap)h(of)g (LP)1962 1987 y Fs(1)2029 1975 y Fj(is)f(at)g(le)l(ast)g Ft(2)18 b Fh(\000)g Fi(\017)29 b Fj(for)i(any)f Fi(\017)23 b(>)f Ft(0)p Fj(.)523 2118 y(Pr)l(o)l(of.)43 b Ft(F)-7 b(or)30 b(a)h(graph)f Fi(G)f Ft(=)f(\()p Fi(V)5 b(;)14 b(E)5 b Ft(\))29 b Fh(2)g Fi(G)p Ft(\()p Fi(\017)p Ft(\),)j(w)n(e)e (assign)g Fi(x)2406 2130 y Fd(ij)2493 2118 y Ft(=)e(1)21 b Fh(\000)f Ft(1)p Fi(=g)32 b Ft(for)f(ev)n(ery)e(edge)523 2218 y(in)k Fi(G)p Ft(,)f(where)g Fi(g)j Ft(is)d(the)h(girth)f(of)g Fi(G)p Ft(.)h(This)f(is)g(a)g(feasible)g(solution)g(for)g(LP)2943 2230 y Fs(1)3012 2218 y Ft(since)g(there)523 2318 y(are)f(no)g(cycles)g (of)h(length)g(less)f(than)h Fi(g)s Ft(.)f(Th)n(us,)g(the)h(optimal)g (solution)f(of)h(LP)3055 2330 y Fs(1)3123 2318 y Ft(has)f(size)523 2417 y(at)26 b(least)f Fh(j)p Fi(E)5 b Fh(j)p Ft(\(1)15 b Fh(\000)g Ft(1)p Fi(=g)s Ft(\).)25 b(The)h(ratio)e(of)i(the)g (optimal)g(fractional)f(solution)g(to)h(the)g(optimal)523 2517 y(in)n(tegral)h(solution)h(is)g(at)g(least)g(\(1)18 b Fh(\000)g Ft(1)p Fi(=g)q Ft(\))p Fi(=)p Ft(\()1938 2484 y Fs(1)p 1938 2498 34 4 v 1938 2546 a(2)1981 2517 y Ft(\(1)h(+)f Fi(\017)p Ft(\)\).)28 b(Since)h Fi(g)d Ft(=)e Fi(\002)r Ft(\()2786 2480 y Fs(log)13 b(log)f Fd(n)p 2787 2498 236 4 v 2835 2546 a Fs(log)h Fd(n)3032 2517 y Ft(\),)29 b(then)g(for)523 2628 y(an)n(y)e Fi(\017)714 2597 y Fc(0)760 2628 y Fi(>)c Ft(0,)k(w)n(e)g(can)g(c)n(ho)r(ose)g Fi(\017)g Ft(and)g Fi(n)h Ft(so)f(that)h(2\(1)17 b Fh(\000)h Ft(1)p Fi(=g)s Ft(\))p Fi(=)p Ft(\(1)f(+)h Fi(\017)p Ft(\))23 b Fh(\025)g Ft(2)18 b Fh(\000)g Fi(\017)3005 2597 y Fc(0)3028 2628 y Ft(.)299 b Fh(u)-55 b(t)648 2771 y Ft(Theorem)26 b(3)i(follo)n(ws)e(from)h(Theorem)g(2)g(and)h(Theorem)f (1.)523 2915 y Fg(Theorem)j(3.)41 b Fj(The)31 b(inte)l(gr)l(ality)f (gap)h(of)g(LP)1962 2927 y Fs(2)2029 2915 y Fj(is)f(at)g(le)l(ast)g Ft(2)18 b Fh(\000)g Fi(\017)29 b Fj(for)i(any)f Fi(\017)23 b(>)f Ft(0)p Fj(.)523 3156 y Fk(3)112 b(Augmen)m(ted)37 b(LP)g(Relaxations)523 3330 y Ft(In)31 b(the)h(previous)e(section,)g(w) n(e)h(sa)n(w)f(that)h(a)g(rather)f(non-trivial)g(LP)g(has)h(an)f(in)n (tegralit)n(y)523 3430 y(gap)25 b(arbitrarily)g(close)g(to)h(2,)g(th)n (us)g(pro)n(viding)f(an)h(upp)r(er)g(b)r(ound)h(that)f(is)g(no)g(b)r (etter)h(than)523 3530 y(the)j(total)f(w)n(eigh)n(t)f(of)i(the)f(edges) g(in)g(the)h(w)n(orst)e(case.)h(Ho)n(w)f(can)h(w)n(e)g(get)g(a)g(b)r (etter)h(upp)r(er)523 3629 y(b)r(ound?)e(One)e(w)n(a)n(y)g(w)n(ould)g (b)r(e)h(to)g(add)g(new)g(constrain)n(ts)e(to)i(this)g(LP)-7 b(.)27 b(Some)f(w)n(ell-kno)n(wn)523 3729 y(constrain)n(ts)19 b(for)h(this)h(problem)f(are)f(the)i(so-called)e Fj(fenc)l(e)j Ft(constrain)n(ts)d(and)h Fj(M\177)-42 b(obius)24 b(ladder)523 3828 y Ft(constrain)n(ts)e(presen)n(ted)h(b)n(y)g(Gr\177)-42 b(otsc)n(hel,)22 b(J)r(\177)-44 b(unger,)23 b(and)g(Reinelt)h([4,)13 b(5].)23 b(In)h(this)f(section,)g(w)n(e)523 3928 y(will)32 b(sho)n(w)f(that)i(that)f(if)h(w)n(e)e(augmen)n(t)h(LP)1923 3940 y Fs(2)1991 3928 y Ft(with)h Fi(k)s Ft(-fence)f(constrain)n(ts)e (for)i(an)n(y)f Fi(k)k Ft(and)523 4028 y(with)29 b Fi(k)s Ft(-M\177)-42 b(obius)27 b(ladder)h(constrain)n(ts)f(for)h Fi(k)f Ft(=)d(3,)k(then)h(the)f(in)n(tegralit)n(y)f(gap)h(remains)f(2.) 523 4127 y(This)f(is)f(also)g(true)h(for)f(5)p Fi(;)14 b Ft(7-ladders,)23 b(but)k(the)f(pro)r(ofs)e(are)h(omitted)h(here.)g(F) -7 b(or)25 b Fi(k)h Fh(\025)c Ft(9,)k(the)523 4227 y(in)n(tegralit)n(y) 19 b(gap)h(of)g(LP)1261 4239 y Fs(2)1318 4227 y Ft(augmen)n(ted)g(with) h Fi(k)s Ft(-M\177)-42 b(obius)20 b(ladder)g(constrain)n(ts)f(is)h(at)h (least)3306 4194 y Fs(33)p 3306 4208 67 4 v 3306 4256 a(17)3382 4227 y Ft(.)523 4327 y(\(Note)k(that)g(the)g(in)n(tegralit)n (y)f(gap)g(of)h(LP)1822 4339 y Fs(1)1883 4327 y Ft(augmen)n(ted)f(with) i(these)e(constrain)n(ts)g(trivially)523 4426 y(remains)34 b(2:)h(a)f(graph)g(b)r(elonging)g(to)h Fi(G)p Ft(\()p Fi(\017)p Ft(\))h(has)e(girth)h(greater)e(than)i(4)g(for)f(su\016cien)n (tly)523 4526 y(small)27 b Fi(\017)h Ft(and)f(therefore)g(do)r(es)g (not)h(con)n(tain)f(an)n(y)f Fi(k)s Ft(-fences)h(or)g Fi(k)s Ft(-M\177)-42 b(obius)27 b(ladders.\))648 4625 y(Throughout)k(this)i(section,)g(it)g(will)g(b)r(e)h(con)n(v)n(enien)n (t)d(to)i(ha)n(v)n(e)f(the)h(follo)n(wing)f(de\014ni-)523 4725 y(tions.)27 b(An)g(edge)g(\()p Fi(i;)14 b(j)5 b Ft(\))27 b(is)g(the)h Fj(c)l(omplementary)g Ft(edge)e(to)h(edge)g(\()p Fi(j;)14 b(i)p Ft(\).)28 b(The)f Fj(value)g Ft(of)g(a)g(set)523 4825 y Fi(S)h Ft(of)c(edges)e(is)i(de\014ned)f(to)h(b)r(e)g Fi(x)p Ft(\()p Fi(S)5 b Ft(\))23 b(=)1756 4762 y Fe(P)1844 4850 y Fs(\()p Fd(i;j)s Fs(\))p Fc(2)p Fd(S)2076 4825 y Fi(x)2123 4837 y Fd(ij)2182 4825 y Ft(.)h(W)-7 b(e)24 b(de\014ne)f(the)h Fj(shortest)g Ft(path)f(from)523 4924 y Fi(i)k Ft(to)h Fi(j)k Ft(in)c(a)f(graph)g Fi(G)h Ft(as)f(a)g(path)h (from)f Fi(i)g Ft(to)h Fi(j)k Ft(with)c(the)g(least)g(total)f(v)-5 b(alue.)p eop %%Page: 6 6 6 5 bop 523 232 a Fq(6)237 b(Alan)n(tha)26 b(Newman)f(and)g(San)n(tosh) g(V)-6 b(empala)523 448 y Fg(3.1)95 b(The)32 b(Bad)g(Example)e(Graph) 523 622 y Ft(W)-7 b(e)32 b(will)g(no)n(w)f(describ)r(e)g(the)h Fj(b)l(ad)i(example)e Ft(graph{the)f(graph)f(whic)n(h)i(w)n(e)f(use)g (to)h(pro)n(v)n(e)523 722 y(our)37 b(lo)n(w)n(er)f(b)r(ound)j(on)e(the) h(in)n(tegralit)n(y)f(gap)g(of)h(the)g(augmen)n(ted)f(LP)g(relaxation.) g(W)-7 b(e)523 821 y(use)36 b(the)h(family)g(of)f(graphs)f Fi(G)p Ft(\()p Fi(\017)p Ft(\))j(de\014ned)f(in)f(Section)h(2.3.)f(W)-7 b(e)36 b(b)r(egin)h(with)g(a)f(graph)523 921 y Fi(G)25 b Ft(=)f(\()p Fi(V)5 b(;)14 b(E)5 b Ft(\))25 b Fh(2)f Fi(G)p Ft(\()p Fi(\017)p Ft(\).)29 b(\(W)-7 b(e)29 b(used)g(this)f (graph)g(to)g(pro)n(v)n(e)f(Theorem)g(2.)h(Ho)n(w)n(ev)n(er,)f(no)n(w)g (w)n(e)523 1020 y(need)c(to)g(assign)f(a)h(v)-5 b(alue)23 b(to)g(ev)n(ery)f(edge)h(in)h(the)f(complete)g(graph.\))g(F)-7 b(or)22 b(ev)n(ery)g(\()p Fi(i;)14 b(j)5 b Ft(\))33 b Fi(=)-52 b Fh(2)24 b Fi(E)5 b Ft(,)523 1120 y(assign)31 b Fi(w)832 1132 y Fd(ij)922 1120 y Ft(=)f(0.)i(F)-7 b(or)31 b(ev)n(ery)g(edge)h(\()p Fi(i;)14 b(j)5 b Ft(\))31 b Fh(2)g Fi(E)5 b Ft(,)32 b(w)n(e)g(assign)f Fi(x)2516 1132 y Fd(ij)2607 1120 y Ft(v)-5 b(alue)32 b(1)21 b Fh(\000)g Ft(1)p Fi(=g)34 b Ft(and)e Fi(x)3346 1132 y Fd(j)s(i)523 1220 y Ft(v)-5 b(alue)27 b(1)p Fi(=g)s Ft(,)e(where)i Fi(g)i Ft(is)e(the)g(girth)f(of)h Fi(G)p Ft(.)g(F)-7 b(or)26 b(all)h Fi(i;)14 b(j)31 b Ft(suc)n(h)c(that)g(neither)g(\()p Fi(i;)14 b(j)5 b Ft(\))26 b(nor)h(\()p Fi(j;)14 b(i)p Ft(\))523 1319 y(are)28 b(in)i Fi(E)5 b Ft(,)29 b(w)n(e)g(assign)e(a)i (v)-5 b(alue)29 b(to)g Fi(x)1687 1331 y Fd(ij)1775 1319 y Ft(using)g(the)g(rule)g(giv)n(en)f(in)i(the)f(pro)r(of)g(of)g (Theorem)523 1419 y(1.)f(W)-7 b(e)28 b(will)g(restate)g(this)g(rule)g (here)f(for)h(the)g(sak)n(e)f(of)h(con)n(v)n(enience.)e(De\014ne)3000 1398 y(\026)2981 1419 y Fi(G)i Ft(to)g(b)r(e)h(the)523 1519 y(graph)20 b(consisting)g(of)i(the)f(edges)g(in)g Fi(E)26 b Ft(and)21 b(their)g(complemen)n(tary)f(edges.)h(Let)g Fi(\013)3103 1531 y Fd(ij)3183 1519 y Ft(b)r(e)h(the)523 1618 y(shortest)29 b(path)h(from)f Fi(i)g Ft(to)h Fi(j)35 b Ft(in)1583 1597 y(\026)1564 1618 y Fi(G)p Ft(.)30 b(De\014ne)g Fi(\013)1994 1630 y Fd(j)s(i)2082 1618 y Ft(similarly)-7 b(.)29 b(Without)i(loss)e(of)g(generalit)n(y)-7 b(,)523 1718 y(assume)27 b Fi(\013)863 1730 y Fd(ij)945 1718 y Fh(\024)22 b Fi(\013)1085 1730 y Fd(j)s(i)1144 1718 y Ft(.)28 b(Then)f(assign)g Fi(x)1704 1730 y Fd(ij)1790 1718 y Ft(and)h Fi(x)1999 1730 y Fd(j)s(i)2085 1718 y Ft(using)g(the)g(follo)n(wing)e(rule.)648 1843 y Fg(Edge)31 b(Assignmen)m(t)e(Rule:)109 b Fi(x)1783 1855 y Fd(ij)1865 1843 y Ft(=)23 b(min)p Fh(f)2143 1810 y Fs(1)p 2143 1824 34 4 v 2143 1871 a(2)2186 1843 y Fi(;)14 b(\013)2276 1855 y Fd(ij)2334 1843 y Fh(g)p Fi(;)97 b(x)2543 1855 y Fd(j)s(i)2625 1843 y Ft(=)22 b(1)c Fh(\000)g Fi(x)2902 1855 y Fd(ij)523 1976 y Ft(The)28 b(follo)n(wing)f(corollary)f(holds)i (for)f(the)i(complete)f(directed)g(graph)2803 1955 y(~)2784 1976 y Fi(G)h Ft(in)f(whic)n(h)g(ev)n(ery)523 2076 y(edge)f(has)g(b)r (een)h(assigned)f(a)g(v)-5 b(alue.)523 2258 y Fg(Corollary)32 b(1.)41 b Fj(If)32 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-1 sc (b) col-1 sh gr /Times-Roman ff 150.00 scf sf 4260 4530 m gs 1 -1 sc (4) col-1 sh gr /Times-Roman ff 150.00 scf sf 2475 3525 m gs 1 -1 sc (1) col-1 sh gr /Times-Roman ff 150.00 scf sf 3045 3525 m gs 1 -1 sc (2) col-1 sh gr /Times-Roman ff 150.00 scf sf 3053 4500 m gs 1 -1 sc (2) col-1 sh gr /Times-Roman ff 150.00 scf sf 2408 4470 m gs 1 -1 sc (1) col-1 sh gr /Times-Roman ff 150.00 scf sf 3683 4492 m gs 1 -1 sc (3) col-1 sh gr /Times-Roman ff 150.00 scf sf 4883 4470 m gs 1 -1 sc (5) col-1 sh gr /Times-Roman ff 150.00 scf sf 3690 3525 m gs 1 -1 sc (3) col-1 sh gr /Times-Roman ff 150.00 scf sf 4252 3518 m gs 1 -1 sc (4) col-1 sh gr /Times-Roman ff 150.00 scf sf 4883 3532 m gs 1 -1 sc (5) col-1 sh gr F2psBeginF2psEnd rs %%EndDocument @endspecial 1951 1027 a currentpoint currentpoint translate 1 0.68481 div 1 0.68481 div scale neg exch neg exch translate 1951 1027 a 1184 1202 a Fo(Fig.)15 b(1.)25 b Fq(A)g(3-ladder)h(\(or)g(3-fence\))g(and)f(a)h(5-ladder.)648 1609 y Ft(Recall)k(that)g(LP)1192 1621 y Fs(2)1259 1609 y Ft(yields)g(an)g(assignmen)n(t)g(for)f(the)i(complete)f(graph.)g (Therefore,)f(in)523 1709 y(order)18 b(to)i(sho)n(w)f(that)g(the)h (total)g(v)-5 b(alue)19 b(of)h(the)g(edges)f(in)h(an)n(y)e Fi(k)s Ft(-ladder)h(is)g(at)h(most)f(3)p Fi(k)5 b Fh(\000)3251 1676 y Fd(k)q Fs(+1)p 3251 1690 121 4 v 3295 1737 a(2)3382 1709 y Ft(,)523 1808 y(it)33 b(su\016ces)g(to)f(sho)n(w)g(that)h(the)g (total)f(v)-5 b(alue)33 b(of)g(the)g(edges)f(in)h(an)n(y)f Fi(k)s Ft(-ladder)f(is)i(at)f(least)533 1875 y Fd(k)q Fs(+1)p 533 1889 V 577 1937 a(2)664 1908 y Ft(.)e(This)g(is)h(b)r (ecause)e(the)i(set)f(of)h(edges)e(complemen)n(tary)g(to)h(a)g Fi(k)s Ft(-ladder)f(also)g(form)h(a)523 2008 y Fi(k)s Ft(-ladder)c(and)i(the)g(sum)g(of)f(the)h(v)-5 b(alues)27 b(of)h(b)r(oth)g Fi(k)s Ft(-ladders)e(is)i(exactly)f(3)p Fi(k)s Ft(.)648 2109 y(No)n(w)d(w)n(e)g(will)g(sho)n(w)g(that)h (constrain)n(t)e(\(1\))h(is)h(satis\014ed)f(for)g(all)g(3-ladders)f(in) 3101 2088 y(~)3082 2109 y Fi(G)p Ft(,)i(whic)n(h)523 2209 y(w)n(e)j(de\014ned)g(in)g(Section)g(3.1.)f(Let)h Fi(M)37 b Ft(b)r(e)28 b(the)h(set)f(of)g(3)p Fi(k)i Ft(edges)d(in)h(a)g Fi(k)s Ft(-ladder)f(and)g(let)i Fi(C)523 2308 y Ft(b)r(e)g(the)g (subset)g(of)f(edges)g(\()p Fi(a)1431 2320 y Fd(i)1459 2308 y Fi(;)14 b(b)1532 2320 y Fd(i)p Fs(+1)1643 2308 y Ft(\))29 b(and)f(\()p Fi(a)1942 2320 y Fd(i)1970 2308 y Fi(;)14 b(b)2043 2320 y Fd(i)p Fc(\000)p Fs(1)2155 2308 y Ft(\))29 b(for)f Fi(i)d Fh(2)g(f)p Ft(1)p Fi(;)14 b(:)g(:)g(:)e(k)s Fh(g)p Ft(.)28 b(The)h(edges)f(in)h Fi(C)523 2408 y Ft(mak)n(e)d(up)h(an)f(undirected)h(cycle)f(of)g(v)-5 b(alue)27 b(2)p Fi(k)s Ft(.)f(F)-7 b(or)26 b(example,)g(in)h(the)g (5-ladder)e(sho)n(wn)h(in)523 2508 y(Figure)j(1,)g Fi(C)36 b Ft(is)29 b(the)h(set)f(of)h(all)f(edges)g(exp)r(ect)g(for)g(the)h(5)f (v)n(ertical)f(edges.)h(F)-7 b(urthermore,)523 2607 y(de\014ne)24 b Fi(T)35 b Ft(to)24 b(b)r(e)g(the)g(set)g(of)g(edges)f(in)1736 2586 y(\026)1717 2607 y Fi(G)i Ft(\(eac)n(h)e(graph)2273 2586 y(~)2254 2607 y Fi(G)h Ft(has)f(a)h(corresp)r(onding)3102 2586 y(\026)3083 2607 y Fi(G)p Ft(,)g(whic)n(h)523 2707 y(is)k(also)f(de\014ned)i(in)g(Section)f(3.1\))f(that)i(b)r(elong)f(to) g(the)h(shortest)e(paths)h(for)g(eac)n(h)f(pair)h(of)523 2806 y(v)n(ertices)33 b Fi(i;)14 b(j)40 b Ft(suc)n(h)34 b(that)h(edge)f(\()p Fi(i;)14 b(j)5 b Ft(\))34 b Fh(2)h Fi(C)6 b Ft(.)35 b(If)g(there)g(are)e(m)n(ultiple)i(shortest)f(paths)h (for)523 2906 y(some)29 b(pair,)g(only)g(one)g(of)h(these)f(paths)h(is) f(included)h(in)g Fi(T)12 b Ft(.)29 b(The)g(edges)g(in)h Fi(T)41 b Ft(will)30 b(pla)n(y)e(a)523 3006 y(k)n(ey)f(role)g(in)g(our) g(pro)r(of.)648 3107 y(W)-7 b(e)20 b(sa)n(y)f(that)i Fi(T)32 b Ft(con)n(tains)19 b(an)h Fj(undir)l(e)l(cte)l(d)g Ft(cycle)g(if)h(the)g(underlying)f(undirected)g(graph)523 3207 y(con)n(tains)f(a)g(cycle.)h(There)f(are)g(t)n(w)n(o)g(cases)g(to) h(consider:)e Fi(T)31 b Ft(can)20 b(either)g(b)r(e)g(a)f(tree)h(or)f (con)n(tain)523 3306 y(an)k(undirected)h(or)e(directed)h(cycle.)h(In)f (the)h(case)e(that)i Fi(T)35 b Ft(con)n(tains)22 b(a)h(cycle,)g(w)n(e)g (will)h(sho)n(w)523 3406 y(that)e(constrain)n(t)f(\(1\))h(is)g (satis\014ed,)g(and)g(when)g Fi(T)33 b Ft(is)22 b(a)g(tree,)g(w)n(e)f (will)i(sho)n(w)e(that)h(constrain)n(t)523 3506 y(\(1\))34 b(is)h(satis\014ed)f(for)f Fi(k)k Ft(=)d(3.)g(Constrain)n(t)f(\(1\))i (is)f(also)f(satis\014ed)h(for)g Fi(k)j Ft(=)c(5)p Fi(;)14 b Ft(7,)34 b(but)h(the)523 3605 y(pro)r(ofs)i(are)f(omitted.)i(Ho)n(w)n (ev)n(er,)d(this)j(constrain)n(t)e(is)i(not)f(necessarily)f (satis\014ed)h(when)523 3705 y Fi(k)f Ft(=)d(9.)h(In)g(other)f(w)n (ords,)f(our)h(edge)g(assignmen)n(t)g(rules)g(could)h(violate)f(some)g Fi(k)s Ft(-ladder)523 3804 y(constrain)n(ts)26 b(for)h Fi(k)f Fh(\025)d Ft(9.)523 4062 y Fg(Case)32 b(1:)f Fa(T)45 b Fg(Con)m(tains)32 b(a)h(Cycle.)41 b Ft(W)-7 b(e)28 b(consider)f(t)n(w)n(o)g(sub)r(cases)g(based)h(on)f(the)h(v)-5 b(alues)523 4162 y(of)31 b(the)g(2)p Fi(k)i Ft(paths)e(in)g Fi(T)42 b Ft(from)30 b Fi(a)1549 4174 y Fd(i)1608 4162 y Ft(to)g Fi(b)1748 4174 y Fd(i)p Fs(+1)1891 4162 y Ft(and)g(from)h Fi(a)2299 4174 y Fd(i)2357 4162 y Ft(to)g Fi(b)2498 4174 y Fd(i)p Fc(\000)p Fs(1)2641 4162 y Ft(for)f Fi(i)e Fh(2)h(f)p Ft(1)p Fi(;)14 b(:)g(:)g(:)e(k)s Fh(g)p Ft(.)31 b(F)-7 b(or)523 4261 y(eac)n(h)33 b(of)h(these)g(cases,)e(w)n(e)i(will)g(sho)n (w)f(that)h(the)g(total)f(v)-5 b(alue)34 b(of)g(the)g(edges)f(in)h Fi(M)42 b Ft(is)34 b(at)523 4361 y(least)727 4328 y Fd(k)q Fs(+1)p 727 4342 V 771 4390 a(2)858 4361 y Ft(.)28 b(In)f(the)h(pro)r (of,)f(w)n(e)h(will)g(use)f(the)h(follo)n(wing)e(lemma.)523 4544 y Fg(Lemma)k(3.)40 b Fj(If)28 b(the)e(value)i(of)f Fi(C)33 b Fj(is)28 b(at)e(le)l(ast)h(1,)h(then)e(the)h(total)g(value)h (of)f(the)g(e)l(dges)g(in)g Fi(M)523 4644 y Fj(is)j(at)g(le)l(ast)911 4611 y Fd(k)q Fs(+1)p 911 4625 V 954 4672 a(2)1041 4644 y Fj(.)523 4825 y(Pr)l(o)l(of.)43 b Ft(If)27 b(the)f(total)g(v)-5 b(alue)25 b(of)h Fi(C)33 b Ft(is)25 b(1)15 b(+)g Fi(\017)25 b Ft(for)h(some)f Fi(\017)e Fh(\025)g Ft(0,)i(then)i(w)n(e)e(will)h (sho)n(w)f(that)i(the)523 4924 y(total)j(v)-5 b(alue)29 b(of)h(the)h(other)e Fi(k)k Ft(edges)c(in)h Fi(M)39 b Ft(is)30 b(at)f(least)2317 4892 y Fd(k)q Fc(\000)p Fs(1)p Fc(\000)p Fd(\017)p 2317 4906 202 4 v 2401 4953 a Fs(2)2528 4924 y Ft(.)h(This)g(w)n(ould)f(imply)i(that)p eop %%Page: 8 8 8 7 bop 523 232 a Fq(8)237 b(Alan)n(tha)26 b(Newman)f(and)g(San)n(tosh) g(V)-6 b(empala)523 448 y Ft(the)29 b(total)f(v)-5 b(alue)29 b(of)f(the)h(edges)f(in)h Fi(M)37 b Ft(is)28 b(\(1)19 b(+)g Fi(\017)p Ft(\))g(+)g(\()2232 416 y Fd(k)q Fc(\000)p Fs(1)p Fc(\000)p Fd(\017)p 2232 430 202 4 v 2316 477 a Fs(2)2443 448 y Ft(\))25 b(=)2599 416 y Fd(k)q Fs(+1+)p Fd(\017)p 2599 430 200 4 v 2682 477 a Fs(2)2837 448 y Ft(as)j(required.)g(W)-7 b(e)523 558 y(just)32 b(need)f(to)g(sho)n(w)g (that)g Fi(x)p Ft(\()p Fi(M)f Fh(n)21 b Fi(C)6 b Ft(\))32 b(is)f(at)g(least)2168 526 y Fd(k)q Fc(\000)p Fs(1)p Fc(\000)p Fd(\017)p 2168 540 202 4 v 2252 587 a Fs(2)2379 558 y Ft(.)g(Consider)g(the)g(equations)g(for)523 658 y(eac)n(h)c(of)h(the)g(directed)g(4-cycles)e(in)i Fi(M)9 b Ft(.)28 b(There)f(are)g Fi(k)k Ft(suc)n(h)c(4-cycles,)g(whic)n(h)h(b) n(y)f(Lemma)523 758 y(1)32 b(eac)n(h)f(ha)n(v)n(e)g(v)-5 b(alue)32 b(at)g(least)f(1.)h(Note)g(that)g(eac)n(h)g(edge)f(in)h Fi(C)39 b Ft(app)r(ears)31 b(in)h(exactly)f(one)523 857 y(equation,)37 b(and)g(eac)n(h)g(edge)f(of)i Fi(M)33 b Fh(n)25 b Fi(C)43 b Ft(app)r(ears)36 b(in)i(exactly)f(t)n(w)n(o)f (equations.)h(Adding)523 957 y(these)23 b Fi(k)j Ft(equations,)d(w)n(e) f(ha)n(v)n(e:)g Fi(x)p Ft(\()p Fi(C)6 b Ft(\))k(+)g(2)p Fi(x)p Ft(\()p Fi(M)19 b Fh(n)10 b Fi(C)c Ft(\))23 b Fh(\025)f Fi(k)s Ft(.)h(By)g(assumption,)g Fi(x)p Ft(\()p Fi(C)6 b Ft(\))24 b(=)f(1)10 b(+)g Fi(\017)p Ft(.)523 1056 y(Therefore,)26 b(w)n(e)i(ha)n(v)n(e:)e Fi(x)p Ft(\()p Fi(M)i Fh(n)18 b Fi(C)6 b Ft(\))24 b Fh(\025)1725 1024 y Fd(k)q Fc(\000)p Fs(1)p Fc(\000)p Fd(\017)p 1725 1038 V 1809 1085 a Fs(2)1936 1056 y Ft(.)1391 b Fh(u)-55 b(t)523 1218 y Fg(Lemma)30 b(4.)40 b Fj(If)31 b Fi(T)40 b Fj(c)l(ontains)30 b(a)g(cycle,)i(then)d(the)h(total)g(value)g(of)h Fi(M)38 b Fj(is)30 b(at)g(le)l(ast)3119 1186 y Fd(k)q Fs(+1)p 3119 1200 121 4 v 3163 1247 a(2)3249 1218 y Fj(.)523 1370 y(Pr)l(o)l(of.)43 b Ft(Note)26 b(that)f(if)h(t)n(w)n(o)f(or)g (more)f(of)i(the)f(2)p Fi(k)j Ft(paths)e(in)f Fi(C)32 b Ft(ha)n(v)n(e)24 b(v)-5 b(alue)26 b(at)f(least)3152 1337 y Fs(1)p 3152 1351 34 4 v 3152 1399 a(2)3195 1370 y Ft(,)g(then)523 1480 y(b)n(y)i(Corollary)f(1)h(and)g(Lemma)h(3,)f (the)h(total)g(v)-5 b(alue)27 b(of)h Fi(M)36 b Ft(is)28 b(at)f(least)2783 1447 y Fd(k)q Fs(+1)p 2783 1461 121 4 v 2827 1509 a(2)2913 1480 y Ft(.)h(No)n(w)f(w)n(e)h(will)523 1580 y(consider)f(t)n(w)n(o)f(remaining)h(cases.)648 1679 y(\(i\))p Fj(A)n(l)t(l)j Ft(2)p Fi(k)i Fj(p)l(aths)f(in)f Fi(T)40 b Fj(have)32 b(value)e(strictly)g(less)g(than)2435 1647 y Fs(1)p 2435 1661 34 4 v 2435 1708 a(2)2478 1679 y Fj(.)e Ft(By)g(Claim)f(1,)g(all)h(edges)f(in)523 1779 y Fi(C)j Ft(ha)n(v)n(e)22 b(v)-5 b(alue)24 b(equal)f(to)h(the)g(v)-5 b(alue)23 b(of)h(their)f(resp)r(ectiv)n(e)g(paths)g(in)h Fi(T)12 b Ft(.)23 b(Since)h(some)f(subset)523 1879 y(of)33 b(the)g(edges)f(in)h Fi(T)45 b Ft(form)32 b(a)h(cycle,)f(the)h(total)g (v)-5 b(alue)33 b(of)g(the)g(edges)f(in)h Fi(T)44 b Ft(is)33 b(at)g(least)f(1.)523 1978 y(Since)f(ev)n(ery)f(edge)h(in)h Fi(T)12 b Ft(,)30 b(b)n(y)h(de\014nition,)h(b)r(elongs)e(to)i(at)f (least)f(one)h(of)h(the)f(2)p Fi(k)j Ft(shortest)523 2078 y(paths,)d(the)h(total)f(v)-5 b(alue)31 b(of)h Fi(C)37 b Ft(is)32 b(at)f(least)g(the)h(total)f(v)-5 b(alue)31 b(of)g Fi(T)12 b Ft(,)31 b(whic)n(h)g(is)g(at)g(least)g(1.)523 2178 y(By)c(Lemma)h(3,)f(the)h(total)f(v)-5 b(alue)28 b(of)f Fi(M)37 b Ft(is)27 b(at)g(least)h(2.)648 2277 y(\(ii\))j Fj(Only)h(one)h(of)g(the)g Ft(2)p Fi(k)i Fj(p)l(aths)d(in)h Fi(T)43 b Fj(has)34 b(value)f(at)f(le)l(ast)2600 2244 y Fs(1)p 2600 2258 V 2600 2306 a(2)2643 2277 y Fj(.)f Ft(W)-7 b(e)31 b(can)f(assume)g(the)523 2377 y(total)e(v)-5 b(alue)28 b(of)g(the)g(other)g(2)p Fi(k)21 b Fh(\000)e Ft(1)27 b(edges)h(in)g Fi(C)34 b Ft(is)28 b(less)g(than)2507 2344 y Fs(1)p 2507 2358 V 2507 2405 a(2)2550 2377 y Ft(.)h(In)f(this)g (case,)g(note)g(that)523 2476 y Fi(T)41 b Ft(do)r(es)30 b(not)g(con)n(tain)f(a)h(directed)g(cycle.)f(If)i(it)f(did,)h(the)f (total)g(v)-5 b(alue)30 b(of)g(the)g(edges)f(in)i Fi(C)523 2576 y Ft(w)n(ould)k(b)r(e)h(at)f(least)h(1.)f(Th)n(us,)g(if)h(w)n(e)f (remo)n(v)n(ed)f(the)i(edge)f(with)h(v)-5 b(alue)35 b(at)h(least)3188 2543 y Fs(1)p 3188 2557 V 3188 2605 a(2)3231 2576 y Ft(,)g(the)523 2676 y(remaining)27 b(edges)g(w)n(ould)g(sum)g(to)h(at)f(least)1958 2643 y Fs(1)p 1958 2657 V 1958 2704 a(2)2001 2676 y Ft(.)648 2775 y(Without)33 b(loss)f(of)g(generalit)n(y)-7 b(,)32 b(assume)g(that)h(edge)f(\()p Fi(a)2401 2787 y Fs(1)2438 2775 y Fi(;)14 b(b)2511 2787 y Fs(2)2548 2775 y Ft(\))33 b(in)g Fi(C)39 b Ft(is)32 b(the)h(edge)f(with)523 2875 y(v)-5 b(alue)20 b(at)h(least)1022 2842 y Fs(1)p 1022 2856 V 1022 2904 a(2)1065 2875 y Ft(.)g(Consider)e(the)i(4-cycle)f Fh(f)p Fi(a)1938 2887 y Fs(1)1974 2875 y Fi(;)14 b(b)2047 2887 y Fs(2)2084 2875 y Fi(;)g(a)2165 2887 y Fs(2)2202 2875 y Fi(;)g(b)2275 2887 y Fs(1)2312 2875 y Fh(g)20 b Ft(in)h Fi(M)29 b Ft(that)21 b(con)n(tains)f(this)g(edge.)523 2975 y(Since)j Fi(T)33 b Ft(con)n(tains)22 b(a)g(path)g(from)g Fi(a)1625 2987 y Fs(2)1685 2975 y Ft(to)g Fi(b)1817 2987 y Fs(1)1877 2975 y Ft(and)g(a)g(path)g(from)h Fi(a)2521 2987 y Fs(1)2580 2975 y Ft(to)f Fi(b)2712 2987 y Fs(2)2749 2975 y Ft(,)h(it)g(cannot)f(con)n(tain)523 3074 y(a)30 b(path)h(from)f Fi(b)1027 3086 y Fs(1)1094 3074 y Ft(to)h Fi(a)1243 3086 y Fs(1)1311 3074 y Ft(and)f(from)g Fi(b)1710 3086 y Fs(2)1778 3074 y Ft(to)g Fi(a)1926 3086 y Fs(2)1963 3074 y Ft(,)h(since)f(then)h(it)g(w)n(ould)f(con)n(tain)g(a)g(directed) 523 3174 y(cycle.)g(Without)g(loss)f(of)h(generalit)n(y)-7 b(,)29 b(assume)g Fi(T)41 b Ft(do)r(es)29 b(not)h(con)n(tain)f(a)h (path)g(from)f Fi(b)3264 3186 y Fs(1)3331 3174 y Ft(to)523 3273 y Fi(a)567 3285 y Fs(1)604 3273 y Ft(.)35 b(Let)f Fi(T)878 3243 y Fc(0)935 3273 y Ft(b)r(e)h(the)g(set)g(of)f(edges)g(in) h Fi(G)g Ft(that)f(corresp)r(ond)f(to)i(the)f(shortest)g(paths)h(for) 523 3373 y(all)e(edges)f(in)h Fi(C)39 b Ft(except)33 b(\()p Fi(a)1413 3385 y Fs(1)1450 3373 y Fi(;)14 b(b)1523 3385 y Fs(2)1560 3373 y Ft(\).)33 b(Then)h Fi(T)1932 3343 y Fc(0)1987 3373 y Ft(con)n(tains)e(a)g(directed)h(or)f (undirected)h(path)523 3473 y(from)g Fi(a)769 3485 y Fs(1)840 3473 y Ft(to)h Fi(b)984 3485 y Fs(1)1021 3473 y Ft(.)g(Since)f(the)h(total)g(v)-5 b(alue)33 b(of)h(the)g(edges)f(in)h Fi(T)2516 3443 y Fc(0)2572 3473 y Ft(is)g(less)f(than)3031 3440 y Fs(1)p 3031 3454 V 3031 3501 a(2)3074 3473 y Ft(,)h(there)f(is) 523 3572 y(a)e(directed)f(or)h(undirected)g(path)g(from)f Fi(a)1877 3584 y Fs(1)1946 3572 y Ft(to)h Fi(b)2087 3584 y Fs(1)2155 3572 y Ft(in)g Fi(G)g Ft(with)h(v)-5 b(alue)30 b(less)h(than)3125 3540 y Fs(1)p 3125 3554 V 3125 3601 a(2)3168 3572 y Ft(.)g(Th)n(us)523 3672 y(the)h(shortest)e(path)i(in)g Fi(G)f Ft(from)g Fi(b)1619 3684 y Fs(1)1688 3672 y Ft(to)g Fi(a)1837 3684 y Fs(1)1905 3672 y Ft(m)n(ust)h(ha)n(v)n(e)e(v)-5 b(alue)31 b(at)h(least)2842 3639 y Fs(1)p 2842 3653 V 2842 3701 a(2)2885 3672 y Ft(.)f(Consider)g(the)533 3749 y Fd(k)q Fc(\000)p Fs(1)p 533 3763 122 4 v 577 3811 a(2)693 3782 y Ft(edge-disjoin)n(t)c(4-cycles)g(in)i Fi(M)37 b Ft(that)28 b(remain)g(when)g(w)n(e)g(remo)n(v)n(e)f(edges)g(with)i (either)523 3882 y(endp)r(oin)n(t)36 b(in)g(the)g(set)g Fh(f)p Fi(a)1357 3894 y Fs(1)1394 3882 y Fi(;)14 b(b)1467 3894 y Fs(1)1504 3882 y Fh(g)p Ft(.)35 b(Then,)h(since)g(edges)f(\()p Fi(b)2362 3894 y Fs(1)2399 3882 y Fi(;)14 b(a)2480 3894 y Fs(1)2517 3882 y Ft(\))36 b(and)g(\()p Fi(a)2831 3894 y Fs(1)2868 3882 y Fi(;)14 b(b)2941 3894 y Fs(2)2978 3882 y Ft(\))36 b(eac)n(h)f(ha)n(v)n(e)523 3981 y(v)-5 b(alue)31 b(at)f(least)1052 3948 y Fs(1)p 1052 3962 34 4 v 1052 4010 a(2)1126 3981 y Ft(and)h(the)g(4-cycle)f(has)g(v)-5 b(alue)30 b(at)h(least)2395 3948 y Fd(k)q Fc(\000)p Fs(1)p 2395 3962 122 4 v 2439 4010 a(2)2527 3981 y Ft(,)g(then)g(the)g(total)f (v)-5 b(alue)31 b(of)523 4091 y(the)d(edges)f(in)h Fi(M)36 b Ft(is)27 b(at)h(least)1491 4058 y Fd(k)q Fs(+1)p 1491 4072 121 4 v 1535 4120 a(2)1622 4091 y Ft(.)1705 b Fh(u)-55 b(t)523 4243 y Fg(Corollary)32 b(2.)41 b Fj(F)-6 b(or)31 b(a)h(3-fenc)l(e)g(in)f(which)i(the)f(c)l(orr)l(esp)l(onding)g Fi(T)43 b Fj(c)l(ontains)31 b(a)h(cycle,)h(ei-)523 4342 y(ther)d Fi(x)p Ft(\()p Fi(C)6 b Ft(\))31 b Fj(is)f(at)g(le)l(ast)g(1,) g(or)g Fi(C)36 b Fj(c)l(ontains)30 b(an)g(e)l(dge)g(of)h(value)f(at)g (le)l(ast)2821 4310 y Fs(1)p 2821 4324 34 4 v 2821 4371 a(2)2864 4342 y Fj(.)523 4573 y Fg(Case)k(2:)f Fa(T)47 b Fg(is)34 b(a)g(T)-8 b(ree.)44 b Ft(W)-7 b(e)30 b(will)g(consider)e(t) n(w)n(o)h(sub)r(cases)g(based)g(on)g(the)h(total)f(v)-5 b(alue)523 4673 y(of)28 b(the)g(edges)e(in)i Fi(T)12 b Ft(.)27 b(F)-7 b(or)27 b(these)h(sub)r(cases,)f(w)n(e)g(will)h(use)f (the)h(follo)n(wing)f(t)n(w)n(o)f(lemmas.)523 4825 y Fg(Lemma)k(5.)40 b Fj(If)29 b Fi(T)39 b Fj(is)29 b(a)f(tr)l(e)l(e,)g (then)g(every)h(e)l(dge)g(in)g Fi(T)39 b Fj(is)28 b(include)l(d)h(in)g (the)f(shortest)g(p)l(aths)523 4924 y(c)l(orr)l(esp)l(onding)j(to)f(at) g(le)l(ast)f(two)h(e)l(dges)h(in)e Fi(C)6 b Fj(.)p eop %%Page: 9 9 9 8 bop 1487 232 a Fq(On)25 b(Relaxations)h(for)h(the)e(Linear)h (Ordering)g(Problem)237 b(9)523 448 y Fj(Pr)l(o)l(of.)43 b Ft(Assume)33 b(there)f(is)g(an)h(edge)f Fi(e)g Ft(in)g Fi(T)44 b Ft(that)33 b(b)r(elongs)f(to)g(only)g(one)g(shortest)g(path) 523 548 y(corresp)r(onding)22 b(to)h(an)g(edge)g(in)h Fi(C)6 b Ft(.)23 b(Consider)g(the)h(set)f(of)g(edges)g(that)h(corresp)r (onds)d(to)j(the)523 648 y(2)p Fi(k)t Fh(\000)q Ft(1)19 b(shortest)f(paths)h(corresp)r(onding)e(to)i(the)g(other)g(2)p Fi(k)t Fh(\000)q Ft(1)f(edges)h(in)g Fi(C)6 b Ft(.)20 b(By)e(assumption,)523 747 y(this)24 b(set)f(of)g(edges)g(do)r(es)g (not)g(include)h(edge)f Fi(e)p Ft(.)g(This)g(set)g(forms)g(a)g (connected)g(graph,)f(since)523 847 y(the)30 b(2)p Fi(k)22 b Fh(\000)e Ft(1)29 b(corresp)r(onding)f(edges)h(form)h(a)f(connected)h (graph.)e(It)j(also)d(con)n(tains)h(all)h(2)p Fi(k)523 946 y Ft(v)n(ertices)37 b(in)i Fi(M)9 b Ft(.)38 b(Th)n(us,)g(if)h(w)n (e)f(add)g(edge)g Fi(e)g Ft(to)g(this)g(graph,)g(it)g(will)h(con)n (tain)e(a)h(cycle,)523 1046 y(implying)28 b(that)g Fi(T)38 b Ft(con)n(tains)27 b(a)g(cycle,)g(whic)n(h)h(is)f(a)h(con)n (tradiction.)700 b Fh(u)-55 b(t)523 1228 y Fg(Lemma)30 b(6.)40 b Fj(If)24 b Fi(T)34 b Fj(is)23 b(a)g(tr)l(e)l(e)g(c)l(orr)l (esp)l(onding)h(to)f(a)g(3-ladder,)i(then)e(for)h(some)f Fi(i)g Fh(2)g(f)p Ft(1)p Fi(;)14 b Ft(2)p Fi(;)g Ft(3)p Fh(g)p Fj(,)523 1328 y Fi(T)41 b Fj(c)l(ontains)30 b(a)g(dir)l(e)l(cte) l(d)g(p)l(ath)h(fr)l(om)f Fi(a)1739 1340 y Fd(i)1796 1328 y Fj(to)g Fi(b)1932 1340 y Fd(i)1989 1328 y Fj(and)g(fr)l(om)g Fi(a)2390 1340 y Fd(i)p Fs(+1)2532 1328 y Fj(to)f Fi(b)2667 1340 y Fd(i)p Fs(+1)2779 1328 y Fj(.)523 1510 y(Pr)l(o)l(of.)43 b Ft(First)34 b(w)n(e)g(will)g(sho)n(w)f(there)g(is)h(a)g(path)g(from)f Fi(a)2326 1522 y Fd(i)2388 1510 y Ft(to)h Fi(b)2532 1522 y Fd(i)2593 1510 y Ft(for)f(some)h Fi(i)p Ft(.)f(Let)i Fi(p)p Ft(\()p Fi(j;)14 b(k)s Ft(\))523 1609 y(denote)28 b(the)g(set)h(of)f(edges)f(in)h(the)h(directed)f(path)g(from)g Fi(j)33 b Ft(to)28 b Fi(k)j Ft(in)d Fi(T)12 b Ft(.)27 b(Since)h Fi(T)40 b Ft(is)28 b(a)f(tree,)523 1709 y(if)33 b Fi(p)p Ft(\()p Fi(j;)14 b(k)s Ft(\))33 b(exists,)g(it)g(is)f(unique.) h(Assume)g(there)f(is)g(not)h(a)f(path)h(from)f Fi(a)2881 1721 y Fd(i)2941 1709 y Ft(to)h Fi(b)3084 1721 y Fd(i)3144 1709 y Ft(for)f(an)n(y)523 1809 y Fi(i)p Ft(.)e(W)-7 b(e)31 b(will)f(sho)n(w)g(that)g(this)h(leads)e(to)i(a)f(con)n (tradiction.)f(Consider)g(the)i(paths)f Fi(p)p Ft(\()p Fi(a)3226 1821 y Fs(1)3263 1809 y Fi(;)14 b(b)3336 1821 y Fs(2)3373 1809 y Ft(\))523 1908 y(and)28 b Fi(p)p Ft(\()p Fi(a)803 1920 y Fs(1)840 1908 y Fi(;)14 b(b)913 1920 y Fs(3)950 1908 y Ft(\).)28 b(The)h(\014rst)e(case)h(is)f(that,)i (without)f(loss)g(of)g(generalit)n(y)-7 b(,)26 b Fi(b)2838 1920 y Fs(2)2903 1908 y Ft(is)i(in)h Fi(p)p Ft(\()p Fi(a)3203 1920 y Fs(1)3240 1908 y Fi(;)14 b(b)3313 1920 y Fs(3)3350 1908 y Ft(\).)523 2008 y(Since)30 b(there)g(m)n(ust)f(b)r(e)i(a)e (directed)h(path)f(from)h Fi(a)2110 2020 y Fs(3)2177 2008 y Ft(to)f Fi(b)2316 2020 y Fs(2)2383 2008 y Ft(and)h(there)f(is)h (a)f(directed)h(path)523 2108 y(from)d Fi(b)755 2120 y Fs(2)820 2108 y Ft(to)h Fi(b)958 2120 y Fs(3)995 2108 y Ft(,)f(there)h(is)f(a)h(directed)f(path)h(from)g Fi(a)2164 2120 y Fs(3)2228 2108 y Ft(to)g Fi(b)2366 2120 y Fs(3)2403 2108 y Ft(.)g(The)g(second)f(case)f(is)i(that)g Fi(b)3368 2120 y Fs(2)523 2207 y Ft(is)g(not)h(in)f Fi(p)p Ft(\()p Fi(a)971 2219 y Fs(1)1008 2207 y Fi(;)14 b(b)1081 2219 y Fs(3)1118 2207 y Ft(\))29 b(and)f Fi(b)1377 2219 y Fs(3)1442 2207 y Ft(is)g(not)h(in)f Fi(p)p Ft(\()p Fi(a)1890 2219 y 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y Ft(,)g(then)g(b)n(y)g(Lemma)f(5)523 1181 y(ev)n(ery)h(edge)g(in)h Fi(T)41 b Ft(b)r(elongs)29 b(to)g(the)i(corresp)r(onding)c(path)j(for)g(at)f(least)h(t)n(w)n(o)f (edges)g(in)h Fi(C)6 b Ft(.)523 1280 y(Th)n(us,)28 b(the)h(total)f(v)-5 b(alue)28 b(of)g Fi(C)35 b Ft(is)28 b(at)g(least)g(1.)g(By)g(Lemma)g(3) g(the)h(total)f(v)-5 b(alue)28 b(of)g Fi(M)37 b Ft(is)28 b(at)523 1380 y(least)727 1347 y Fd(k)q Fs(+1)p 727 1361 121 4 v 771 1408 a(2)858 1380 y Ft(.)648 1479 y(If)36 b(the)g(total)f(v)-5 b(alue)36 b(of)f(the)h(edges)f(in)h Fi(T)47 b Ft(is)35 b(less)g(than)2467 1447 y Fs(1)p 2467 1461 34 4 v 2467 1508 a(2)2510 1479 y Ft(,)h(then)g(b)n(y)g(Lemma)f(6)g (and)523 1579 y(without)28 b(loss)f(of)g(generalit)n(y)-7 b(,)26 b(assume)h Fi(T)39 b Ft(con)n(tains)26 b(a)h(directed)h(path)f (from)h Fi(a)3005 1591 y Fs(1)3069 1579 y Ft(to)g Fi(b)3207 1591 y Fs(1)3271 1579 y Ft(and)523 1679 y(from)k Fi(a)768 1691 y Fs(2)838 1679 y Ft(to)h Fi(b)981 1691 y Fs(2)1018 1679 y Ft(.)f(Let)h Fi(s)g Ft(b)r(e)g(the)g(v)-5 b(alue)32 b(of)h(the)g(directed)f(path)h(from)f Fi(a)2799 1691 y Fs(1)2869 1679 y Ft(to)h Fi(b)3012 1691 y Fs(1)3081 1679 y Ft(in)g Fi(T)12 b Ft(,)32 b(i.e.)523 1778 y Fi(s)d Ft(=)684 1716 y Fe(P)771 1803 y Fd(ij)s Fc(2)p Fd(p)p Fs(\()p Fd(a)965 1811 y Fb(1)999 1803 y Fd(;b)1048 1811 y Fb(1)1080 1803 y Fs(\))1124 1778 y Fi(x)1171 1790 y Fd(ij)1230 1778 y Ft(.)i(Note)g(that)g Fi(s)e(<)1842 1746 y Fs(1)p 1842 1760 V 1842 1807 a(2)1916 1778 y Ft(since)i(the)g (total)g(v)-5 b(alue)31 b(of)f(the)i(edges)e(in)h Fi(T)42 b Ft(is)523 1896 y(less)23 b(than)871 1863 y Fs(1)p 871 1877 V 871 1925 a(2)914 1896 y Ft(.)h(The)f(v)-5 b(alue)23 b(of)h Fi(x)1475 1908 y Fd(b)1504 1916 y Fb(1)1537 1908 y Fd(a)1573 1916 y Fb(1)1632 1896 y Ft(is)g(at)f(least)g(1)10 b Fh(\000)g Fi(s)22 b Ft(and)h(eac)n(h)f(edge)h(in)h(the)f(path)h(from) f Fi(a)3368 1908 y Fs(1)523 1996 y Ft(to)h Fi(b)657 2008 y Fs(1)719 1996 y Ft(is)g(on)g(a)g(path)g(from)g Fi(a)1404 2008 y Fs(1)1466 1996 y Ft(to)g Fi(b)1600 2008 y Fs(2)1661 1996 y Ft(or)g(from)g Fi(a)1997 2008 y Fs(2)2058 1996 y Ft(to)g Fi(b)2192 2008 y Fs(1)2229 1996 y Ft(,)h(so)e(the)i(sum)g(of) f Fi(x)2826 2008 y Fd(a)2862 2016 y Fb(1)2895 2008 y Fd(b)2924 2016 y Fb(2)2985 1996 y Ft(and)g Fi(x)3190 2008 y Fd(a)3226 2016 y Fb(2)3259 2008 y Fd(b)3288 2016 y Fb(1)3349 1996 y Ft(is)523 2095 y(at)i(least)g Fi(s)p Ft(.)g(Therefore,)g(the)g(total)g(v)-5 b(alue)26 b(of)h(the)f(edges)g (\()p Fi(b)2378 2107 y Fs(1)2415 2095 y Fi(;)14 b(a)2496 2107 y Fs(1)2533 2095 y Ft(\))p Fi(;)g Ft(\()p Fi(a)2678 2107 y Fs(1)2716 2095 y Fi(;)g(b)2789 2107 y Fs(2)2825 2095 y Ft(\))p Fi(;)27 b Ft(and)f(\()p Fi(a)3143 2107 y Fs(2)3181 2095 y Fi(;)14 b(b)3254 2107 y Fs(1)3290 2095 y Ft(\))27 b(is)523 2195 y(at)e(least)g(1.)f(The)h(4)g(edges)f(in) i Fi(M)33 b Ft(that)26 b(ha)n(v)n(e)e(neither)h(endp)r(oin)n(t)g(in)g (the)h(set)f Fh(f)p Fi(a)3003 2207 y Fs(1)3039 2195 y Fi(;)14 b(b)3112 2207 y Fs(1)3149 2195 y Fh(g)25 b Ft(mak)n(e)523 2295 y(up)i(a)f(directed)h(4-cycle,)e(so)h(they)h(ha)n(v)n(e)e(total)i (v)-5 b(alue)26 b(at)h(least)f(1.)g(Th)n(us)h(an)n(y)e(3-ladder)h(has) 523 2394 y(v)-5 b(alue)28 b(at)f(least)g(2.)3350 2494 y Fh(u)-55 b(t)648 2594 y Ft(F)-7 b(or)30 b Fi(k)h Ft(=)d(5)p Fi(;)14 b Ft(7,)29 b(it)i(is)g(also)f(the)h(case)e(that)i(the)h(total)e (v)-5 b(alue)31 b(of)f(the)h(edges)f(in)h Fi(M)40 b Ft(is)30 b(at)523 2693 y(least)723 2660 y Fd(k)q Fs(+1)p 723 2674 121 4 v 767 2722 a(2)854 2693 y Ft(.)23 b(These)g(pro)r(ofs)f(are)g (omitted)i(here.)f(F)-7 b(or)22 b Fi(k)k Ft(=)d(9,)g(it)g(is)g(not)h (necessarily)d(the)j(case)523 2793 y(that)33 b(the)g(total)g(sum)g(of)g (the)g(edges)f(in)h(a)g(M\177)-42 b(obius)32 b(ladder)h(is)f(at)h (least)g(5.)f(W)-7 b(e)33 b(conclude)523 2892 y(this)28 b(section)f(with)h(a)f(b)r(ound)h(for)f(the)h(general)f(case.)523 3059 y Fg(Theorem)j(4.)41 b Fj(The)23 b(inte)l(gr)l(ality)f(gap)g(of)g (LP)1928 3071 y Fs(2)1987 3059 y Fj(augmente)l(d)f(by)h(M\177)-42 b(obius)22 b(ladder)h(c)l(onstr)l(aints)523 3159 y(is)30 b(at)g(le)l(ast)911 3126 y Fs(33)p 911 3140 67 4 v 911 3188 a(17)1005 3159 y Fh(\000)18 b Fi(\017)30 b Fj(for)g(any)g Fi(\017)23 b(>)g Ft(0)p Fj(.)523 3326 y(Pr)l(o)l(of.)43 b Ft(When)25 b Fi(k)i Ft(is)d(at)g(least)f(9,)h(then)g Fi(T)36 b Ft(is)23 b(a)h(tree)g(with)g(at)g(least)g(18)f(unique)h(v)n (ertices)f(and)523 3425 y(at)i(least)h(17)e(edges.)h(If)h(ev)n(ery)e (edge)h(original)f(edge)h(in)h Fi(G)g Ft(is)f(assigned)g(v)-5 b(alue)2952 3393 y Fs(33)p 2952 3407 V 2952 3454 a(34)3054 3425 y Ft(and)25 b(ev)n(ery)523 3525 y(complemen)n(tary)32 b(edge)h(is)g(assigned)f(v)-5 b(alue)1975 3492 y Fs(1)p 1959 3506 V 1959 3554 a(34)2035 3525 y Ft(,)33 b(then)h(the)g(total)f (v)-5 b(alue)33 b(of)g(the)h(edges)f(in)523 3625 y Fi(T)40 b Ft(is)29 b(at)g(least)1006 3592 y Fs(1)p 1006 3606 34 4 v 1006 3653 a(2)1049 3625 y Ft(.)g(Th)n(us,)g(the)g(ob)5 b(jectiv)n(e)29 b(v)-5 b(alue)29 b(of)g(the)g(LP)g(relaxation)e(augmen) n(ted)h(with)523 3724 y(constrain)n(t)23 b(\(1\))h(for)g(all)g Fi(k)j Ft(will)e(b)r(e)f(at)g(least)g(33)p Fh(j)p Fi(E)5 b Fh(j)p Fi(=)p Ft(34.)22 b(Since)j(the)f(optimal)g(in)n(tegral)f(v)-5 b(alue)523 3824 y(is)34 b(arbitrarily)f(close)h(to)g Fh(j)p Fi(E)5 b Fh(j)p Fi(=)p Ft(2)34 b(for)g(su\016cien)n(tly)g(large) g Fi(n)p Ft(,)g(the)h(in)n(tegralit)n(y)e(gap)h(of)g(LP)3368 3836 y Fs(2)523 3923 y Ft(extended)28 b(b)n(y)f(M\177)-42 b(obius)28 b(ladder)e(constrain)n(ts)h(is)g(at)h(least)2351 3891 y Fs(33)p 2351 3905 67 4 v 2351 3952 a(17)2445 3923 y Fh(\000)18 b Fi(\017)28 b Ft(for)f(an)n(y)g Fi(\017)22 b(>)h Ft(0.)267 b Fh(u)-55 b(t)523 4167 y Fg(3.3)95 b(F)-8 b(ence)31 b(Constrain)m(ts)523 4327 y Ft(A)e Fi(k)s Fj(-fenc)l(e)g Ft(is)g(obtained)g(b)n(y)g(directing)f(the)h(edges)g(of)g(a)f(complete) h(undirected)g(bipartite)523 4426 y(graph)d(on)h(2)p Fi(k)j Ft(v)n(ertices)c(as)g(follo)n(ws:)g(the)i(2)p Fi(k)i Ft(v)n(ertices)c(are)g(divided)h(in)n(to)g(t)n(w)n(o)g(sets,)g Fi(A)g Ft(and)523 4526 y Fi(B)t Ft(,)f(of)g Fi(k)j Ft(v)n(ertices)c (eac)n(h.)g(Eac)n(h)g(v)n(ertex)g(from)g(set)h Fi(A)g Ft(is)g(paired)f(with)i(a)e(v)n(ertex)g(from)h(set)g Fi(B)523 4625 y Ft(and)f(eac)n(h)f(of)h(these)g(pairs)g(is)g(connected) f(with)i(a)f Fj(up)g Ft(edge.)g(All)g(other)g(edges)f(are)g(directed) 523 4725 y Fj(down)p Ft(.)k(A)g(3-fence)f(is)h(sho)n(wn)f(in)h(Figure)f (1.)648 4825 y(An)f(acyclic)f(subgraph)f(of)i(a)f(3-fence)g(includes)h (at)g(most)f(7)h(of)f(the)i(9)e(edges.)g(Ho)n(w)n(ev)n(er,)523 4924 y(there)f(is)g(a)g(fractional)f(solution)h(of)g(7)1700 4892 y Fs(1)p 1700 4906 34 4 v 1700 4953 a(2)1767 4924 y Ft(that)h(satis\014es)e(LP)2355 4936 y Fs(2)2392 4924 y Ft(:)h(eac)n(h)g(do)n(wn)f(edge)h(is)g(assigned)p eop %%Page: 11 11 11 10 bop 1487 232 a Fq(On)25 b(Relaxations)h(for)h(the)e(Linear)h (Ordering)g(Problem)198 b(11)523 448 y Ft(a)25 b(v)-5 b(alue)26 b(of)f(1)h(and)f(eac)n(h)g(up)h(edge)f(is)h(assigned)2030 416 y Fs(1)p 2030 430 34 4 v 2030 477 a(2)2073 448 y Ft(.)g(In)g(general,)e(an)h(acyclic)g(subgraph)g(of)g(a)523 548 y Fi(k)s Ft(-fence)i(includes)g(at)g(most)f Fi(k)1473 518 y Fs(2)1528 548 y Fh(\000)16 b Fi(k)k Ft(+)d(1)26 b(of)h(the)h(edges.)e(Ho)n(w)n(ev)n(er,)f(w)n(e)h(can)h(alw)n(a)n(ys)e (\014nd)i(a)523 648 y(fractional)f(solution)h(with)h(v)-5 b(alue)27 b Fi(k)1657 617 y Fs(2)1711 648 y Fh(\000)1804 615 y Fd(k)p 1804 629 37 4 v 1806 676 a Fs(2)1877 648 y Ft(that)h(satis\014es)e(ev)n(ery)g(cycle)h(constrain)n(t.)f(Th)n(us,) 523 747 y(if)d(w)n(e)f(add)g(fence)h(constrain)n(ts)e(to)h(the)h(LP)f (relaxation,)f(w)n(e)h(ma)n(y)g(get)g(a)g(fractional)g(solution)523 847 y(that)32 b(is)f(a)g(b)r(etter)g(appro)n(ximation)f(of)h(the)h(in)n (tegral)e(solution.)h(F)-7 b(ence)31 b(constrain)n(ts)f(state)523 946 y(that)e(the)g(total)f(v)-5 b(alue)28 b(of)f(the)h(edges)f(in)h(an) n(y)f Fi(k)s Ft(-fence)g(cannot)g(exceed)g Fi(k)2838 916 y Fs(2)2894 946 y Fh(\000)18 b Fi(k)j Ft(+)d(1.)1638 1075 y Fe(X)1482 1257 y Fs(\()p Fd(i;j)s Fs(\))p Fc(2)p Fd(k)q Fc(\000)p Fd(f)7 b(ence)1810 1154 y Fi(x)1857 1166 y Fd(ij)1939 1154 y Fh(\024)22 b Fi(k)2072 1120 y Fs(2)2128 1154 y Fh(\000)c Fi(k)j Ft(+)d(1)899 b(\(2\))648 1396 y(W)-7 b(e)29 b(will)f(sho)n(w)g(that)h(despite)g(the)g(fact)f (that)h(LP)2218 1408 y Fs(2)2284 1396 y Ft(is)f(strengthened)g(b)n(y)h (adding)f(con-)523 1495 y(strain)n(t)k(\(2\),)h(the)g(in)n(tegralit)n (y)e(gap)g(of)i(this)g(augmen)n(ted)f(LP)g(relaxation)f(remains)g(2.)h (T)-7 b(o)523 1595 y(sho)n(w)34 b(this,)i(w)n(e)f(will)h(again)e(use)h (the)g(graph)2016 1574 y(~)1998 1595 y Fi(G)g Ft(discussed)g(in)h (Section)f(3.1.)f(Since)i(a)f(3-)523 1695 y(ladder)30 b(is)g(also)f(a)g(3-fence,)h(w)n(e)g(kno)n(w)f(that)i(the)f(total)g(v) -5 b(alue)30 b(of)g(an)n(y)g(3-fence)f(is)h(at)g(most)523 1794 y(7)i(in)h(this)f(solution.)g(Th)n(us,)g(w)n(e)g(can)g(sho)n(w)f (that)i(the)g(total)f(v)-5 b(alue)32 b(of)g(an)n(y)g Fi(k)s Ft(-fence)g(is)g(at)523 1894 y(most)27 b Fi(k)772 1864 y Fs(2)828 1894 y Fh(\000)18 b Fi(k)j Ft(+)d(1.)523 2064 y Fg(Lemma)30 b(8.)40 b Fj(The)31 b(total)f(value)g(of)h(any)f Fi(k)s Fj(-fenc)l(e)g(is)g(at)g(most)f Fi(k)2533 2034 y Fs(2)2589 2064 y Fh(\000)18 b Fi(k)j Ft(+)d(1)p Fj(.)523 2234 y(Pr)l(o)l(of.)43 b Ft(The)38 b(set)h(of)f(complemen)n(tary)f (edges)g(of)i(a)e Fi(k)s Ft(-fence)h(also)f(form)h(a)g Fi(k)s Ft(-fence)g(and)523 2334 y(hence)25 b(\(2\))g(is)f(equiv)-5 b(alen)n(t)25 b(to)f(the)i(condition)e(that)h(the)g(total)g(v)-5 b(alue)24 b(of)h(edges)f(in)h(a)f Fi(k)s Ft(-fence)523 2434 y(is)g Fj(at)j(le)l(ast)e Fi(k)15 b Fh(\000)d Ft(1.)23 b(In)i(Lemma)f(7,)g(w)n(e)g(sho)n(w)n(ed)f(that)i(the)g(lemma)f(is)g (true)h(for)e Fi(k)j Ft(=)d(3,)h(whic)n(h)523 2533 y(will)g(b)r(e)g (the)g(base)f(case)f(for)h(our)g(inductiv)n(e)h(pro)r(of.)f(W)-7 b(e)24 b(will)f(assume)g(that)h(the)g(total)f(v)-5 b(alue)523 2633 y(of)25 b(an)n(y)f(\()p Fi(k)16 b Fh(\000)d Ft(1\)-fence)23 b(is)i(at)f(least)h Fi(k)15 b Fh(\000)e Ft(2)24 b(and)h(sho)n(w)e(that) i(the)h(total)e(v)-5 b(alue)25 b(of)f(an)n(y)g Fi(k)s Ft(-fence)523 2733 y(is)k(at)f(least)g Fi(k)21 b Fh(\000)d Ft(1.)648 2832 y(A)k Fi(k)s Ft(-fence)f(con)n(tains)1329 2765 y Fe(\000)1367 2795 y Fd(k)1369 2861 y Fs(3)1404 2765 y Fe(\001)1464 2832 y Ft(distinct)h(3-ladders)e(\(or)i(3-fences\)) f(as)g(subgraphs.)f(F)-7 b(or)22 b(some)523 2932 y(3-ladder)i(con)n (tained)h(in)h(a)f Fi(k)s Ft(-fence,)g(if)h(the)g(corresp)r(onding)e Fi(T)36 b Ft(is)26 b(a)f(tree)g(\()p Fi(T)37 b Ft(and)25 b Fi(C)32 b Ft(corre-)523 3031 y(sp)r(onding)22 b(to)h(a)f(3-ladder)f (are)h(de\014ned)h(in)g(Section)g(3.2\))f(with)h(v)-5 b(alue)22 b(less)h(than)3044 2999 y Fs(1)p 3044 3013 34 4 v 3044 3060 a(2)3087 3031 y Ft(,)g(then)g(b)n(y)523 3131 y(Lemma)29 b(6)f(for)h(some)f Fi(i)d Fh(2)g(f)p Ft(1)p Fi(;)14 b Ft(2)p Fi(;)g Ft(3)p Fh(g)p Ft(,)27 b(there)h(is)h(a)g(directed)f(path)i(from)e Fi(a)2822 3143 y Fd(i)2879 3131 y Ft(to)g Fi(b)3017 3143 y Fd(i)3074 3131 y Ft(and)g(from)523 3231 y Fi(a)567 3243 y Fd(i)p Fs(+1)708 3231 y Ft(to)i Fi(b)848 3243 y Fd(i)p Fs(+1)959 3231 y Ft(.)g(Th)n(us,)f(the)h(total)g(v)-5 b(alue)29 b(of)h(the)g(edges)f(\()p Fi(b)2343 3243 y Fd(i)2370 3231 y Fi(;)14 b(a)2451 3243 y Fd(i)2479 3231 y Ft(\))p Fi(;)g Ft(\()p Fi(a)2624 3243 y Fd(i)2652 3231 y Fi(;)g(b)2725 3243 y Fd(i)p Fs(+1)2836 3231 y Ft(\),)30 b(and)g(\()p Fi(a)3161 3243 y Fd(i)p Fs(+1)3272 3231 y Fi(;)14 b(b)3345 3243 y Fd(i)3373 3231 y Ft(\))523 3330 y(is)28 b(at)g(least)g(1.)g (When)g(w)n(e)g(remo)n(v)n(e)e(all)i(edges)g(that)g(ha)n(v)n(e)f(one)h (endp)r(oin)n(t)g(in)g Fh(f)p Fi(a)3039 3342 y Fd(i)3066 3330 y Fi(;)14 b(b)3139 3342 y Fd(i)3167 3330 y Fh(g)27 b Ft(from)523 3430 y(the)j Fi(k)s Ft(-fence,)f(w)n(e)g(are)f(left)i (with)g(a)f(\()p Fi(k)23 b Fh(\000)c Ft(1\)-fence.)29 b(By)g(induction,)h(this)g(fence)f(has)g(v)-5 b(alue)523 3530 y(at)28 b(least)f Fi(k)21 b Fh(\000)d Ft(2.)27 b(Th)n(us,)h(the)g (total)f(v)-5 b(alue)27 b(of)h(the)g Fi(k)s Ft(-fence)f(is)h(at)f (least)g Fi(k)22 b Fh(\000)c Ft(1.)648 3629 y(If)31 b(it)h(is)f(the)g (case)f(that)i(for)e(none)h(of)g(the)h(3-ladders)d(con)n(tained)i(in)g (the)h Fi(k)s Ft(-fence,)f(the)523 3729 y(corresp)r(onding)21 b Fi(T)33 b Ft(is)23 b(a)f(tree)g(with)h(v)-5 b(alue)23 b(less)f(than)2179 3696 y Fs(1)p 2179 3710 V 2179 3757 a(2)2222 3729 y Ft(,)h(then)g(w)n(e)f(will)h(sho)n(w)e(that)i(the)g (total)523 3828 y(v)-5 b(alue)33 b(of)h(the)g(edges)e(directed)i(from)f Fi(A)g Ft(to)h Fi(B)j Ft(in)d(the)g Fi(k)s Ft(-fence)f(is)h(at)f(least) g(1.)g(Consider)523 3928 y(a)e(particular)f(3-ladder)g(that)i(is)f(a)g (subgraph)f(of)i(the)g Fi(k)s Ft(-fence.)f(If)h(the)f(corresp)r(onding) f Fi(T)523 4028 y Ft(is)k(a)f(tree)g(with)h(v)-5 b(alue)34 b(at)f(least)1593 3995 y Fs(1)p 1593 4009 V 1593 4056 a(2)1636 4028 y Ft(,)h(then)g(this)g(is)g(true)f(b)n(y)g(Lemma)h(5.)f (If)h Fi(T)45 b Ft(con)n(tains)32 b(a)523 4127 y(cycle,)25 b(then)h(b)n(y)f(Corollary)e(2,)h(the)i(set)f(of)h(edges)e Fi(C)32 b Ft(corresp)r(onding)23 b(to)i(this)h(3-ladder)e(has)523 4227 y(v)-5 b(alue)24 b(at)h(least)f(1)g(or)g(con)n(tains)f(an)i(edge)f Fi(e)g Ft(with)h(at)g(least)2359 4194 y Fs(1)p 2359 4208 V 2359 4256 a(2)2402 4227 y Ft(.)f(In)h(the)g(latter)f(case,)g (consider)523 4327 y(another)c(3-ladder)g(subgraph)g(that)h(do)r(es)f (not)i(con)n(tain)e(edge)g Fi(e)p Ft(.)h(Using)g(Corollary)e(2)h (again,)523 4426 y(one)29 b(of)g(the)h(edges)e(in)i Fi(C)35 b Ft(corresp)r(onding)28 b(to)h Fj(this)h Ft(3-ladder)d(m)n(ust)j(also) e(con)n(tain)h(an)f(edge)523 4526 y(with)f(v)-5 b(alue)26 b(at)f(least)1226 4493 y Fs(1)p 1226 4507 V 1226 4554 a(2)1269 4526 y Ft(.)h(Th)n(us,)g(the)h(total)e(v)-5 b(alue)26 b(of)g(the)h(edges)e(directed)h(from)g Fi(A)g Ft(to)g Fi(B)k Ft(is)523 4625 y(at)e(least)f(1.)648 4725 y(Let)k Fi(Y)50 b Ft(b)r(e)32 b(the)f(set)h(of)f(edges)f(from)h Fi(b)1854 4737 y Fd(i)1913 4725 y Ft(to)g Fi(a)2062 4737 y Fd(i)2121 4725 y Ft(and)g(let)g Fi(X)38 b Ft(b)r(e)32 b(the)f(set)h(of)f(edges)f(from)523 4825 y Fi(a)567 4837 y Fd(i)626 4825 y Ft(to)h Fi(b)767 4837 y Fd(j)802 4825 y Ft(.)g(If)h(w)n(e)f(consider)f(the)i Fi(k)i Ft(p)r(ossible)d(\()p Fi(k)24 b Fh(\000)d Ft(1\)-fences)30 b(that)i(are)e(subgraphs)g(of)i (the)523 4924 y Fi(k)s Ft(-fence,)37 b(w)n(e)g(see)g(that)g(eac)n(h)f (edge)h(in)h Fi(Y)55 b Ft(is)37 b(used)h(in)f Fi(k)28 b Fh(\000)c Ft(1)37 b(of)g(these)g(\()p Fi(k)28 b Fh(\000)c Ft(1\)-fences,)p eop %%Page: 12 12 12 11 bop 523 232 a Fq(12)199 b(Alan)n(tha)26 b(Newman)f(and)g(San)n (tosh)g(V)-6 b(empala)523 448 y Ft(and)33 b(eac)n(h)f(edge)h(in)g Fi(X)39 b Ft(is)33 b(used)g(in)g Fi(k)25 b Fh(\000)d Ft(2)32 b(of)h(these)g(\()p Fi(k)25 b Fh(\000)d Ft(1\)-fences.)33 b(By)f(the)i(induction)523 548 y(h)n(yp)r(othesis,)27 b(eac)n(h)g(\()p Fi(k)21 b Fh(\000)c Ft(1\)-fence)27 b(has)g(v)-5 b(alue)27 b(at)g(least)g Fi(k)21 b Fh(\000)d Ft(2)27 b(b)n(y)g(induction.)h(Com)n(bining)523 648 y(these)34 b Fi(k)j Ft(equations,)c(w)n(e)g(ha)n(v)n(e:)g(\()p Fi(k)25 b Fh(\000)e Ft(1\))p Fi(x)p Ft(\()p Fi(Y)c Ft(\))k(+)f(\()p Fi(k)k Fh(\000)c Ft(2\))p Fi(x)p Ft(\()p Fi(X)7 b Ft(\))33 b Fh(\025)g Fi(k)s Ft(\()p Fi(k)26 b Fh(\000)c Ft(2\).)34 b(W)-7 b(e)34 b(kno)n(w)523 747 y(that)f Fi(x)p Ft(\()p Fi(X)7 b Ft(\))32 b(is)h(at)f(least)g(1.)g(The)g(minim)n(um)h(v)-5 b(alue)33 b(of)f Fi(x)p Ft(\()p Fi(X)7 b Ft(\))22 b(+)f Fi(x)p Ft(\()p Fi(Y)f Ft(\))32 b(that)h(satis\014es)f(the)523 847 y(equation)e(is)g Fi(x)p Ft(\()p Fi(X)7 b Ft(\))20 b(+)g Fi(x)p Ft(\()p Fi(Y)g Ft(\))27 b(=)g Fi(k)c Fh(\000)d Ft(1,)30 b(i.e.)g Fi(x)p Ft(\()p Fi(X)7 b Ft(\))28 b(=)f(1)p Fi(;)14 b(x)p Ft(\()p Fi(Y)19 b Ft(\))28 b(=)f Fi(k)c Fh(\000)d Ft(2.)30 b(Th)n(us,)g(the)g(total)523 946 y(v)-5 b(alue)29 b(of)h(the)g(edges)e(in)i(a)f Fi(k)s Ft(-fence)g(is)h(at)f (least)g Fi(k)22 b Fh(\000)e Ft(1,)29 b(whic)n(h)g(implies)h(that)g(it) f(is)h(also)e(at)523 1046 y(most)f Fi(k)772 1016 y Fs(2)828 1046 y Fh(\000)18 b Fi(k)j Ft(+)d(1.)2227 b Fh(u)-55 b(t)523 1215 y Fg(Theorem)30 b(5.)41 b Fj(The)g(inte)l(gr)l(ality)g (gap)f(of)h(LP)2002 1227 y Fs(2)2080 1215 y Fj(augmente)l(d)e(with)i (fenc)l(e)f(c)l(onstr)l(aints)g(is)523 1315 y Ft(2)18 b Fh(\000)g Fi(\017)29 b Fj(for)i(any)f Fi(\017)23 b(>)g Ft(0)p Fj(.)523 1576 y Fk(4)112 b(Lo)m(w)m(er)38 b(Bounds)g(on)f(Appro) m(ximation)523 1771 y Ft(In)26 b(this)f(section,)g(w)n(e)g(describ)r(e) g(a)g(reduction)h(from)f(the)g(problem)g(of)h(\014nding)f(a)g(maxim)n (um)523 1871 y(satis\014able)35 b(subset)g(of)h(a)f(giv)n(en)g(set)h (of)f(linear)g(equations)g(mo)r(dulo)g(2)g(with)i(three)e(v)-5 b(ari-)523 1970 y(ables)23 b(p)r(er)g(equation)f(to)h(the)h(maxim)n(um) f(acyclic)g(subgraph)f(problem.)g(F)-7 b(or)23 b(this)g(problem,)523 2070 y(H)-10 b(\027)-52 b(astad)27 b(pro)n(v)n(ed)f(the)i(follo)n(wing) f(tigh)n(t)g(b)r(ound.)523 2239 y Fg(Theorem)j(6)42 b(\(H)-13 b(\027)-59 b(astad)31 b([7]\).)f Fj(F)-6 b(or)29 b(every)h Fi(\017)23 b(>)g Ft(0)p Fj(,)29 b(it)g(is)h(NP-har)l(d)g(to)f(tel)t(l)h (if)g(a)f(given)h(set)523 2339 y(of)25 b(line)l(ar)f(e)l(quations)g(mo) l(dulo)g(2)g(with)h(thr)l(e)l(e)e(variables)j(is)e(satis\014able)h(or)f (at)g(most)f Fi(m)p Ft(\()3220 2306 y Fs(1)p 3221 2320 34 4 v 3221 2367 a(2)3269 2339 y Ft(+)5 b Fi(\017)p Ft(\))523 2438 y Fj(of)31 b(its)e(clauses)i(ar)l(e)f(satis\014able.)648 2607 y Ft(W)-7 b(e)23 b(use)h(Theorem)e(6)h(and)g(the)h(reduction)f (describ)r(ed)g(b)r(elo)n(w)g(to)g(obtain)g(the)h(follo)n(wing)523 2707 y(lo)n(w)n(er)30 b(b)r(ound)h(on)g(the)h(maxim)n(um)f(acyclic)f (subgraph)g(problem)h(\(and)g(hence)g(the)h(linear)523 2807 y(ordering)26 b(problem\).)523 2976 y Fg(Theorem)k(7.)41 b Fj(It)d(is)g(NP-har)l(d)g(to)g(appr)l(oximate)h(the)f(maximum)g (acyclic)h(sub)l(gr)l(aph)g(to)523 3075 y(within)785 3043 y Fs(65)p 785 3057 67 4 v 785 3104 a(66)880 3075 y Ft(+)18 b Fi(\017)30 b Fj(for)g(any)g Fi(\017)23 b(>)g Ft(0)p Fj(.)648 3244 y Ft(Giv)n(en)34 b(a)h(set)g(of)g Fi(m)g Ft(linear)f(equations)g(on)h Fi(n)g Ft(v)-5 b(ariables,)33 b(w)n(e)i(construct)f(a)h(graph)f Fi(G)523 3344 y Ft(using)24 b(the)i(follo)n(wing)d(rules:)i(\(w)n(e)f(assume)g(all)h(equations)f (ha)n(v)n(e)f(the)j(righ)n(t)e(hand)g(side)h(zero)523 3444 y(b)n(y)i(negating)g(one)g(literal)g(if)h(necessary)-7 b(.\))558 3620 y(1.)41 b(F)-7 b(or)27 b(eac)n(h)g(v)-5 b(ariable)26 b Fi(x)e Fh(2)f Fi(F)12 b Ft(,)27 b(w)n(e)g(create)g(t)n (w)n(o)g(v)n(ertices)f(and)h(t)n(w)n(o)g(edges.)g(The)g(v)n(ertices)664 3720 y(are)34 b Fi(x)857 3732 y Fs(0)930 3720 y Ft(and)g Fi(x)1145 3732 y Fs(1)1218 3720 y Ft(and)h(the)g(edges)f(are)g(\()p Fi(x)1991 3732 y Fs(0)2029 3720 y Fi(;)14 b(x)2113 3732 y Fs(1)2151 3720 y Ft(\))35 b(and)g(\()p Fi(x)2466 3732 y Fs(1)2504 3720 y Fi(;)14 b(x)2588 3732 y Fs(0)2625 3720 y Ft(\).)36 b(These)e(v)n(ertices)g(and)664 3819 y(edges)27 b(will)h(form)f(the)h Fj(variable)i Ft(gadget)c(for)h Fi(x)p Ft(.)558 3918 y(2.)41 b(F)-7 b(or)30 b(eac)n(h)f(clause)h Fi(C)1312 3930 y Fd(j)1375 3918 y Fh(2)d Fi(F)12 b Ft(,)31 b(w)n(e)f(construct)f(a)h Fj(clause)h Ft(gadget.)e(Eac)n(h)g(clause)h (has)g(the)664 4017 y(form)c Fi(x)17 b Ft(+)f Fi(y)j Ft(+)d Fi(z)26 b Fh(\021)d Ft(0)j(where)g Fi(x;)14 b(y)29 b Ft(and)e Fi(z)j Ft(are)25 b(literals.)h(F)-7 b(or)26 b(a)g(literal)g Fi(x)h Ft(in)g(the)g(clause)664 4117 y(w)n(e)d(create)g(a)g(4-cycle)g Fh(f)p Fi(x)1452 4129 y Fs(2)1489 4117 y Fi(;)14 b(x)1573 4129 y Fs(3)1610 4117 y Fi(;)g(x)1694 4129 y Fs(4)1732 4117 y Fi(;)g(x)1816 4129 y Fs(5)1854 4117 y Fh(g)p Ft(.)24 b(W)-7 b(e)25 b(lab)r(el)f(edge)g(\()p Fi(x)2545 4129 y Fs(5)2583 4117 y Fi(;)14 b(x)2667 4129 y Fs(2)2705 4117 y Ft(\))25 b(as)f Fi(x)f Ft(=)g(1)h(and)g(edge)664 4217 y(\()p Fi(x)743 4229 y Fs(3)781 4217 y Fi(;)14 b(x)865 4229 y Fs(4)903 4217 y Ft(\))29 b(as)f Fi(x)d Ft(=)f(0.)k(W)-7 b(e)29 b(also)f(do)g(this)h(for)f(the)h(literals)f Fi(y)j Ft(and)d Fi(z)t Ft(.)g(Then)h(w)n(e)f(add)h(the)664 4316 y(follo)n(wing)i(12)f (edges)h(as)g(sho)n(wn)g(in)g(Figure)g(3:)g(\()p Fi(z)2254 4328 y Fs(2)2291 4316 y Fi(;)14 b(x)2375 4328 y Fs(5)2413 4316 y Ft(\))p Fi(;)32 b Ft(\()p Fi(z)2571 4328 y Fs(2)2608 4316 y Fi(;)14 b(y)2686 4328 y Fs(3)2723 4316 y Ft(\))p Fi(;)32 b Ft(\()p Fi(z)2881 4328 y Fs(4)2918 4316 y Fi(;)14 b(y)2996 4328 y Fs(3)3033 4316 y Ft(\))p Fi(;)32 b Ft(\()p Fi(z)3191 4328 y Fs(4)3228 4316 y Fi(;)14 b(x)3312 4328 y Fs(3)3350 4316 y Ft(\))p Fi(;)664 4416 y Ft(\()p Fi(x)743 4428 y Fs(2)781 4416 y Fi(;)g(z)857 4428 y Fs(3)894 4416 y Ft(\))p Fi(;)28 b Ft(\()p Fi(x)1056 4428 y Fs(2)1094 4416 y Fi(;)14 b(y)1172 4428 y Fs(5)1209 4416 y Ft(\))p Fi(;)28 b Ft(\()p Fi(x)1371 4428 y Fs(4)1409 4416 y Fi(;)14 b(z)1485 4428 y Fs(5)1522 4416 y Ft(\))p Fi(;)28 b Ft(\()p Fi(x)1684 4428 y Fs(4)1722 4416 y Fi(;)14 b(y)1800 4428 y Fs(5)1836 4416 y Ft(\))p Fi(;)28 b Ft(\()p Fi(y)1992 4428 y Fs(2)2030 4416 y Fi(;)14 b(z)2106 4428 y Fs(3)2142 4416 y Ft(\))p Fi(;)28 b Ft(\()p Fi(y)2298 4428 y Fs(2)2336 4416 y Fi(;)14 b(z)2412 4428 y Fs(5)2448 4416 y Ft(\))p Fi(;)28 b Ft(\()p Fi(y)2604 4428 y Fs(4)2642 4416 y Fi(;)14 b(x)2726 4428 y Fs(3)2763 4416 y Ft(\))p Fi(;)28 b Ft(\()p Fi(y)2919 4428 y Fs(4)2956 4416 y Fi(;)14 b(x)3040 4428 y Fs(5)3078 4416 y Ft(\).)558 4515 y(3.)41 b(Eac)n(h)27 b(clause)f(gadget)h(is)g(link)n(ed)h(to)f(the)h(appropriate)e(v)-5 b(ariable)27 b(gadgets)f(as)h(follo)n(ws.)741 4627 y Fh(\001)41 b Ft(F)-7 b(or)22 b(a)h(literal)f Fi(x)p Ft(,)h(w)n(e)f (connect)h(the)g(corresp)r(onding)e(4-cycle)g(in)i(the)g(clause)f (gadget)805 4726 y(to)f(the)f(v)-5 b(ariable)20 b(gadget)f(b)n(y)h (adding)g(edges)g(\()p Fi(x)2267 4738 y Fs(2)2305 4726 y Fi(;)14 b(x)2389 4738 y Fs(1)2426 4726 y Ft(\))p Fi(;)g Ft(\()p Fi(x)2574 4738 y Fs(1)2612 4726 y Fi(;)g(x)2696 4738 y Fs(3)2734 4726 y Ft(\))p Fi(;)g Ft(\()p Fi(x)2882 4738 y Fs(0)2920 4726 y Fi(;)g(x)3004 4738 y Fs(5)3042 4726 y Ft(\))p Fi(;)g Ft(\()p Fi(x)3190 4738 y Fs(4)3228 4726 y Fi(;)g(x)3312 4738 y Fs(0)3350 4726 y Ft(\).)741 4825 y Fh(\001)41 b Ft(F)-7 b(or)22 b(a)h(literal)p 1249 4779 48 4 v 22 w Fi(x)p Ft(,)g(w)n(e)f(connect)h(the)g(corresp)r (onding)e(4-cycle)g(in)i(the)g(clause)f(gadget)805 4924 y(to)f(the)f(v)-5 b(ariable)20 b(gadget)f(b)n(y)h(adding)g(edges)g(\()p Fi(x)2267 4936 y Fs(2)2305 4924 y Fi(;)14 b(x)2389 4936 y Fs(0)2426 4924 y Ft(\))p Fi(;)g Ft(\()p Fi(x)2574 4936 y Fs(0)2612 4924 y Fi(;)g(x)2696 4936 y Fs(3)2734 4924 y Ft(\))p Fi(;)g Ft(\()p Fi(x)2882 4936 y Fs(1)2920 4924 y Fi(;)g(x)3004 4936 y Fs(5)3042 4924 y Ft(\))p Fi(;)g Ft(\()p Fi(x)3190 4936 y Fs(4)3228 4924 y Fi(;)g(x)3312 4936 y Fs(1)3350 4924 y Ft(\).)p eop %%Page: 13 13 13 12 bop 1487 232 a Fq(On)25 b(Relaxations)h(for)h(the)e(Linear)h (Ordering)g(Problem)198 b(13)648 448 y Ft(The)28 b(resulting)h(graph)e Fi(G)i Ft(has)g(36)p Fi(m)18 b Ft(+)h(2)p Fi(n)28 b Ft(edges,)g(36)g (edges)g(for)g(eac)n(h)g(clause)g(gadget)523 548 y(and)i(t)n(w)n(o)f (edges)g(for)h(eac)n(h)f(v)-5 b(ariable)29 b(gadget.)g(In)h(order)e(to) i(relate)f(v)-5 b(ariable)29 b(assignmen)n(ts)523 648 y(to)34 b(acyclic)f(subgraphs)g(of)h Fi(G)p Ft(,)g(w)n(e)g(sa)n(y)f (that)h(remo)n(ving)e(edge)i(\()p Fi(x)2636 660 y Fs(1)2674 648 y Fi(;)14 b(x)2758 660 y Fs(0)2795 648 y Ft(\))35 b(\(lab)r(eled)f Fi(x)g Ft(=)f(1)523 747 y(in)h(Figure)f(3\))h(corresp) r(onds)d(to)j(setting)g(the)g(v)-5 b(ariable)32 b Fi(x)j Ft(to)e(true,)h(and)f(remo)n(ving)f(edge)523 847 y(\()p Fi(x)602 859 y Fs(0)640 847 y Fi(;)14 b(x)724 859 y Fs(1)762 847 y Ft(\))33 b(\(lab)r(eled)g Fi(x)f Ft(=)f(0)i(in)g(Figure)f(3\))g (corresp)r(onds)f(to)i(setting)g(v)-5 b(ariable)31 b Fi(x)j Ft(to)e(false.)523 946 y(Throughout)40 b(the)h(pro)r(of,)g(w)n (e)g(will)g(refer)f(to)h(edges)f(lab)r(eled)i Fi(x)k Ft(=)f(0)40 b(or)g Fi(x)46 b Ft(=)f(1)c(in)g(a)523 1046 y(clause)35 b(gadget)f(for)g(a)h(literal)g Fi(x)h Ft(\(i.e)f(edges)g (\()p Fi(x)2052 1058 y Fs(5)2090 1046 y Fi(;)14 b(x)2174 1058 y Fs(2)2211 1046 y Ft(\))36 b(and)f(\()p Fi(x)2527 1058 y Fs(3)2565 1046 y Fi(;)14 b(x)2649 1058 y Fs(4)2687 1046 y Ft(\)\))36 b(and)f(the)g(edges)g(in)523 1146 y(the)30 b(v)-5 b(ariable)28 b(gadgets)h(as)g Fj(lab)l(ele)l(d)i 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setgray ef gr col0 s % Polyline gs clippath 2300 4854 m 2355 4965 l 2255 4893 l 2342 4996 l 2388 4957 l cp clip n 2400 885 m 2399 886 l 2398 888 l 2395 892 l 2390 898 l 2384 907 l 2375 918 l 2363 934 l 2349 952 l 2332 975 l 2312 1002 l 2289 1033 l 2263 1068 l 2234 1106 l 2202 1149 l 2168 1196 l 2131 1246 l 2092 1299 l 2051 1355 l 2009 1414 l 1965 1475 l 1920 1538 l 1874 1603 l 1828 1668 l 1782 1734 l 1736 1801 l 1691 1867 l 1647 1934 l 1603 2000 l 1561 2064 l 1521 2128 l 1482 2191 l 1445 2253 l 1410 2313 l 1377 2371 l 1347 2428 l 1318 2483 l 1293 2536 l 1269 2588 l 1248 2639 l 1230 2688 l 1214 2735 l 1201 2781 l 1190 2827 l 1181 2871 l 1175 2914 l 1172 2957 l 1170 3000 l 1171 3044 l 1174 3089 l 1179 3133 l 1188 3179 l 1198 3225 l 1212 3272 l 1228 3320 l 1247 3369 l 1269 3419 l 1293 3471 l 1320 3524 l 1349 3579 l 1381 3635 l 1415 3692 l 1452 3751 l 1490 3811 l 1531 3873 l 1573 3935 l 1616 3998 l 1661 4061 l 1707 4125 l 1753 4189 l 1800 4252 l 1846 4314 l 1892 4376 l 1938 4435 l 1982 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l 3699 3513 l 3687 3498 l 3677 3486 l 3670 3478 l 3660 3465 l gs col0 s gr gr % arrowhead n 3709 3578 m 3660 3465 l 3757 3542 l 3733 3560 l 3709 3578 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 3710 4856 m 3630 4950 l 3657 4829 l 3596 4950 l 3650 4977 l cp clip n 3615 3465 m 3616 3467 l 3618 3470 l 3621 3476 l 3626 3486 l 3633 3499 l 3642 3517 l 3653 3539 l 3667 3565 l 3682 3595 l 3698 3628 l 3716 3663 l 3735 3701 l 3754 3741 l 3773 3781 l 3792 3821 l 3811 3861 l 3828 3900 l 3844 3938 l 3859 3975 l 3872 4009 l 3883 4043 l 3893 4074 l 3901 4105 l 3907 4134 l 3911 4161 l 3914 4188 l 3915 4215 l 3914 4242 l 3912 4268 l 3909 4296 l 3903 4325 l 3896 4354 l 3887 4386 l 3876 4418 l 3863 4453 l 3849 4488 l 3834 4526 l 3818 4564 l 3800 4603 l 3782 4642 l 3764 4681 l 3745 4720 l 3727 4757 l 3710 4792 l 3694 4824 l 3680 4853 l 3667 4878 l 3656 4899 l 3647 4917 l 3640 4930 l 3630 4950 l gs col0 s gr gr % arrowhead n 3710 4856 m 3630 4950 l 3657 4829 l 3684 4843 l 3710 4856 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 2239 2502 m 2340 2430 l 2285 2541 l 2373 2438 l 2327 2399 l cp clip n 2370 5985 m 2369 5984 l 2368 5982 l 2364 5979 l 2359 5973 l 2352 5965 l 2343 5954 l 2330 5939 l 2315 5922 l 2296 5900 l 2275 5875 l 2250 5847 l 2222 5814 l 2192 5778 l 2158 5739 l 2122 5696 l 2083 5651 l 2042 5602 l 2000 5551 l 1956 5498 l 1911 5443 l 1865 5387 l 1819 5330 l 1772 5272 l 1726 5214 l 1681 5155 l 1636 5097 l 1593 5040 l 1551 4983 l 1510 4927 l 1472 4872 l 1436 4818 l 1401 4766 l 1369 4715 l 1340 4666 l 1313 4618 l 1288 4571 l 1266 4526 l 1247 4482 l 1231 4439 l 1217 4398 l 1205 4357 l 1196 4317 l 1190 4278 l 1186 4239 l 1185 4200 l 1186 4160 l 1189 4119 l 1195 4078 l 1204 4036 l 1216 3993 l 1231 3950 l 1248 3905 l 1268 3859 l 1291 3811 l 1317 3762 l 1346 3712 l 1377 3660 l 1411 3606 l 1447 3551 l 1486 3495 l 1527 3437 l 1569 3378 l 1613 3319 l 1659 3259 l 1705 3199 l 1752 3138 l 1800 3078 l 1847 3019 l 1894 2961 l 1940 2904 l 1985 2850 l 2028 2798 l 2069 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l gs col0 s gr gr % arrowhead n 3646 5847 m 3630 5970 l 3586 5854 l 3616 5851 l 3646 5847 l cp gs 0.00 setgray ef gr col0 s /Times-Roman ff 180.00 scf sf 9675 2550 m gs 1 -1 sc (x) col0 sh gr /Times-Italic ff 180.00 scf sf 9675 1050 m gs 1 -1 sc (x) col0 sh gr /Times-Italic ff 180.00 scf sf 9600 3525 m gs 1 -1 sc (y) col0 sh gr /Times-Italic ff 180.00 scf sf 9600 4950 m gs 1 -1 sc (y) col0 sh gr /Times-Italic ff 180.00 scf sf 9675 5925 m gs 1 -1 sc (z) col0 sh gr /Times-Italic ff 180.00 scf sf 9675 7425 m gs 1 -1 sc (z) col0 sh gr /Times-Roman ff 150.00 scf sf 9810 2610 m gs 1 -1 sc (0) col0 sh gr /Times-Roman ff 150.00 scf sf 9705 4980 m gs 1 -1 sc (0) col0 sh gr /Times-Roman ff 150.00 scf sf 9795 7485 m gs 1 -1 sc (0) col0 sh gr /Times-Roman ff 150.00 scf sf 9795 1095 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 150.00 scf sf 9705 3615 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 150.00 scf sf 9795 5985 m gs 1 -1 sc (1) col0 sh gr /Times-Italic ff 180.00 scf sf 2460 1155 m gs 1 -1 sc (x) col0 sh gr /Times-Italic ff 180.00 scf sf 2505 2205 m gs 1 -1 sc (x) col0 sh gr /Times-Italic ff 180.00 scf sf 2490 6225 m gs 1 -1 sc (z) col0 sh gr /Times-Italic ff 180.00 scf sf 2415 7335 m gs 1 -1 sc (z) col0 sh gr /Times-Italic ff 180.00 scf sf 3420 2175 m gs 1 -1 sc (x) col0 sh gr /Times-Roman ff 150.00 scf sf 3540 2235 m gs 1 -1 sc (4) col0 sh gr /Times-Roman ff 150.00 scf sf 2595 1200 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 150.00 scf sf 2610 2235 m gs 1 -1 sc (5) col0 sh gr /Times-Italic ff 180.00 scf sf 3420 1155 m gs 1 -1 sc (x) col0 sh gr /Times-Roman ff 150.00 scf sf 3525 1185 m gs 1 -1 sc (3) col0 sh gr /Times-Roman ff 150.00 scf sf 2595 6285 m gs 1 -1 sc (2) col0 sh gr /Times-Roman ff 150.00 scf sf 2520 7395 m gs 1 -1 sc (5) col0 sh gr /Times-Roman ff 150.00 scf sf 3442 4853 m gs 1 -1 sc (4) col0 sh gr /Times-Italic ff 180.00 scf sf 3374 4793 m gs 1 -1 sc (y) col0 sh gr /Times-Italic ff 180.00 scf sf 3480 6240 m gs 1 -1 sc (z) col0 sh gr /Times-Roman ff 150.00 scf sf 3585 6300 m gs 1 -1 sc (3) col0 sh gr /Times-Italic ff 180.00 scf sf 3420 7320 m gs 1 -1 sc (z) col0 sh gr /Times-Roman ff 150.00 scf sf 3495 7395 m gs 1 -1 sc (4) col0 sh gr /Times-Italic ff 180.00 scf sf 2565 4800 m gs 1 -1 sc (y) col0 sh gr /Times-Roman ff 150.00 scf sf 2655 4875 m gs 1 -1 sc (5) col0 sh gr /Times-Italic ff 180.00 scf sf 8790 1800 m gs 1 -1 sc (x=0) col0 sh gr /Times-Italic ff 180.00 scf sf 10155 1800 m gs 1 -1 sc (x=1) col0 sh gr /Times-Italic ff 180.00 scf sf 8760 4245 m gs 1 -1 sc (y=0) col0 sh gr /Times-Italic ff 180.00 scf sf 10125 4215 m gs 1 -1 sc (y=1) col0 sh gr /Times-Italic ff 180.00 scf sf 10140 6660 m gs 1 -1 sc (z=1) col0 sh gr /Times-Italic ff 180.00 scf sf 8805 6690 m gs 1 -1 sc (z=0) col0 sh gr /Times-Italic ff 180.00 scf sf 3435 1665 m gs 1 -1 sc (x=0) col0 sh gr /Times-Italic ff 180.00 scf sf 3480 6840 m gs 1 -1 sc (z=0) col0 sh gr /Times-Italic ff 180.00 scf sf 2325 1650 m gs 1 -1 sc (x=1) col0 sh gr /Times-Italic ff 180.00 scf sf 2310 6825 m gs 1 -1 sc (z=1) col0 sh gr /Times-Italic ff 180.00 scf sf 2348 3735 m gs 1 -1 sc (y) col0 sh gr /Times-Roman ff 150.00 scf sf 2445 3825 m gs 1 -1 sc (2) col0 sh gr /Times-Italic ff 180.00 scf sf 3577 3720 m gs 1 -1 sc (y) col0 sh gr /Times-Roman ff 150.00 scf sf 3667 3795 m gs 1 -1 sc (3) col0 sh gr /Times-Italic ff 180.00 scf sf 2220 4245 m gs 1 -1 sc (y=1) col0 sh gr /Times-Italic ff 180.00 scf sf 3555 4245 m gs 1 -1 sc (y=0) col0 sh gr F2psBegin10setmiterlimitn−10008850m−1000−1000l11462−1000l114628850lcpclip0.049800.04980scF2psEnd rs %%EndDocument @endspecial 739 3312 a currentpoint currentpoint translate 1 0.58557 div 1 0.58557 div scale neg exch neg exch translate 739 3312 a 990 3486 a Fo(Fig.)15 b(3.)25 b Fq(The)h(clause)g(and)g(v)l(ariable)g(gadgets)h(for)f Fn(x)17 b Fq(+)g Fn(y)i Fq(+)e Fn(z)24 b Fl(\021)d Fq(0.)523 3945 y Fg(Lemma)30 b(9.)40 b Fj(A)30 b(minimal)h(fe)l(e)l(db)l(ack)g (ar)l(c)f(set)f(is)h(acyclic.)523 4129 y(Pr)l(o)l(of.)43 b Ft(F)-7 b(or)31 b(an)n(y)h(acyclic)f(graph,)f(there)i(is)g(an)f (ordering)g(of)h(the)g(v)n(ertices)f(suc)n(h)g(that)h(all)523 4229 y(edges)e(in)g(the)h(graph)e(are)g(forw)n(ard)g(edges,)h(i.e.)g (for)g(an)g(edge)g(\()p Fi(i;)14 b(j)5 b Ft(\),)30 b Fi(i)g Ft(precedes)g Fi(j)35 b Ft(in)c(the)523 4328 y(ordering.)36 b(Giv)n(en)i(a)g(feedbac)n(k)f(arc)g(set,)h(consider)f(suc)n(h)h(an)g (ordering)e(for)h(the)i(acyclic)523 4428 y(graph)f(that)i(is)f (obtained)h(b)n(y)f(deleting)g(the)h(feedbac)n(k)f(arc)f(set.)i(If)g (an)n(y)e(edges)h(in)h(the)523 4527 y(feedbac)n(k)35 b(arc)f(set)h(are)f(forw)n(ard)f(edges,)i(then)g(the)h(feedbac)n(k)e (arc)g(set)i(is)f(not)g(minimal,)523 4627 y(since)27 b(suc)n(h)g(edges)f(can)g(b)r(e)i(added)f(to)g(the)g(acyclic)f(graph)g (without)h(creating)f(an)n(y)h(cycles.)523 4727 y(Th)n(us,)35 b(the)h(feedbac)n(k)f(arc)f(set)i(m)n(ust)f(consist)g(only)g(of)g(bac)n (kw)n(ard)f(edges)g(and)i(hence)f(is)523 4826 y(itself)28 b(acyclic.)2360 b Fh(u)-55 b(t)p eop %%Page: 14 14 14 13 bop 523 232 a Fq(14)199 b(Alan)n(tha)26 b(Newman)f(and)g(San)n (tosh)g(V)-6 b(empala)523 448 y Fg(Lemma)30 b(10.)40 b Fj(Ther)l(e)25 b(is)f(a)g(minimum)f(fe)l(e)l(db)l(ack)j(ar)l(c)e(set) f(of)i Fi(G)f Fj(c)l(onsisting)g(of)g(only)h(lab)l(ele)l(d)523 548 y(e)l(dges.)523 740 y(Pr)l(o)l(of.)43 b Ft(Note)36 b(that)f(ev)n(ery)f(cycle)h(in)h Fi(G)g Ft(con)n(tains)e(lab)r(eled)h (edges.)g(This)g(is)h(b)r(ecause)f(for)523 840 y(ev)n(ery)23 b(non-lab)r(eled)h(edge)f(\()p Fi(i;)14 b(j)5 b Ft(\))25 b(in)f Fi(G)p Ft(,)h Fi(i)f Ft(is)g(a)g(v)n(ertex)f(suc)n(h)h(that)g (the)h(only)f(incoming)f(edge)523 940 y(is)h(a)f(lab)r(eled)h(edge,)g (and)f Fi(j)29 b Ft(is)24 b(a)g(v)n(ertex)e(suc)n(h)i(that)g(the)g (only)g(outgoing)f(edges)g(is)g(a)h(lab)r(eled)523 1039 y(edge.)g(Th)n(us,)h(giv)n(en)f(a)g(feedbac)n(k)g(arc)g(set)h Fi(F)37 b Ft(con)n(taining)23 b(a)i(non-lab)r(eled)f(edge)g(in)h(a)g (clause)523 1139 y(gadget,)34 b(w)n(e)g(can)g(\014nd)h(another)f (feedbac)n(k)g(arc)f(set)i Fi(F)2287 1109 y Fc(0)2345 1139 y Ft(with)g Fh(j)p Fi(F)2629 1109 y Fc(0)2652 1139 y Fh(j)g(\024)f(j)p Fi(F)12 b Fh(j)35 b Ft(b)n(y)f(replacing)523 1238 y(eac)n(h)27 b(non-lab)r(eled)g(edge)g(with)h(either)g(of)f(its)h (adjacen)n(t)f(lab)r(eled)h(edges.)537 b Fh(u)-55 b(t)523 1431 y Fg(Lemma)30 b(11.)40 b Fj(The)26 b(minimum)f(fe)l(e)l(db)l(ack)h (ar)l(c)g(set)e(for)i(the)f(gr)l(aph)h Fi(G)g Fj(c)l(ontains)f Fi(n)8 b Ft(+)g(3)p Fi(m)g Ft(+)g Fi(u)523 1530 y Fj(e)l(dges,)31 b(wher)l(e)f Fi(u)g Fj(is)g(the)g(minimum)f(numb)l(er)g(of)i (unsatis\014e)l(d)e(e)l(quations.)523 1723 y(Pr)l(o)l(of.)43 b Ft(By)27 b(Lemma)g(9,)f(exactly)g(one)h(edge)f(from)h(ev)n(ery)f(v)-5 b(ariable)26 b(gadget)g(is)g(in)i(the)f(min-)523 1822 y(im)n(um)h(feedbac)n(k)f(arc)g(set.)g(In)h(addition,)g(w)n(e)f(sho)n (w)f(the)i(follo)n(wing)f(things:)648 1927 y(\(i\))c(F)-7 b(or)22 b(a)g(clause,)g Fi(x)9 b Ft(+)g Fi(y)j Ft(+)d Fi(z)25 b Ft(=)d(0,)h(and)f(an)h(assignmen)n(t)e(that)i(satis\014es)f (this)h(clause,)f(w)n(e)523 2026 y(need)32 b(to)f(remo)n(v)n(e)f(only)h (three)g(edges)g(from)g(the)h(corresp)r(onding)e(clause)g(gadget)h(so)g (that)523 2126 y(the)22 b(subgraph)f(consisting)g(of)h(the)g(clause)g (gadget)e(and)i(its)g(three)g(corresp)r(onding)e(v)-5 b(ariable)523 2225 y(gadgets)22 b(is)i(acyclic.)f(By)g(Lemma)g(10,)g(w) n(e)g(only)h(need)f(to)h(remo)n(v)n(e)e(lab)r(eled)h(edges)g(from)h (the)523 2325 y(clause)31 b(gadget.)g(There)g(are)g(four)g(satisfying)g (assignmen)n(ts)g(for)g(the)h(v)-5 b(ariables)31 b Fi(x;)14 b(y)s(;)g(z)35 b Ft(in)523 2425 y(this)d(clause.)g(They)g(are:)f Fh(ff)p Ft(0)p Fi(;)14 b Ft(0)p Fi(;)g Ft(0)p Fh(g)p Fi(;)29 b Fh(f)p Ft(0)p Fi(;)14 b Ft(1)p Fi(;)g Ft(1)p Fh(g)p Fi(;)30 b Fh(f)p Ft(1)p Fi(;)14 b Ft(1)p Fi(;)g Ft(0)p Fh(g)p Fi(;)29 b Fh(f)p Ft(1)p Fi(;)14 b Ft(0)p Fi(;)g Ft(1)p Fh(gg)p Ft(.)29 b(F)-7 b(or)32 b(eac)n(h)f(of)h(these)523 2524 y(four)26 b(assignmen)n(ts,)f(w)n(e)g(remo)n(v)n(e)g(the)h(three)g (edges)f(with)i(the)g(opp)r(osite)e(assignmen)n(t)g(from)523 2624 y(the)32 b(clause)e(gadget.)g(F)-7 b(or)31 b(example,)g(for)g(the) g(assignmen)n(t)f Fi(x)g Ft(=)f Fi(y)i Ft(=)e Fi(z)j Ft(=)d(0,)i(w)n(e)g(remo)n(v)n(e)523 2724 y(the)d(edges)f(lab)r(eled)h Fi(x)23 b Ft(=)g(1)p Fi(;)14 b(y)25 b Ft(=)d(1)p Fi(;)14 b(z)26 b Ft(=)d(1.)648 2828 y(\(ii\))34 b(F)-7 b(or)33 b(a)g(clause,)g Fi(x)23 b Ft(+)f Fi(y)j Ft(+)d Fi(z)36 b Ft(=)d(0,)g(and)g(an)n(y)g(assignmen)n(t)g(that)h(do)r(es)f(not)g (satisfy)523 2927 y(this)28 b(clause,)f(w)n(e)g(need)g(to)h(remo)n(v)n (e)d(four)i(edges)g(from)g(the)h(corresp)r(onding)d(clause)i(gadget)523 3027 y(so)d(that)h(the)h(subgraph)d(consisting)h(of)h(the)h(clause)e (gadget)g(and)g(its)h(three)g(corresp)r(onding)523 3127 y(v)-5 b(ariable)34 b(gadgets)g(is)i(acyclic.)e(There)h(are)g(four)g (assignmen)n(ts)f(to)h(the)h(v)-5 b(ariables)34 b Fi(x;)14 b(y)s(;)g(z)523 3226 y Ft(that)27 b(do)f(not)h(satisfy)f(this)h (clause.)g(They)f(are:)g Fh(ff)p Ft(1)p Fi(;)14 b Ft(1)p Fi(;)g Ft(1)p Fh(g)p Fi(;)24 b Fh(f)p Ft(0)p Fi(;)14 b Ft(0)p Fi(;)g Ft(1)p Fh(g)p Fi(;)24 b Fh(f)p Ft(1)p Fi(;)14 b Ft(0)p Fi(;)g Ft(0)p Fh(g)p Fi(;)24 b Fh(f)p Ft(0)p Fi(;)14 b Ft(1)p Fi(;)g Ft(0)p Fh(gg)p Ft(.)523 3326 y(F)-7 b(or)22 b(eac)n(h)h(of)g(these)g(four)g(assignmen)n(ts,)e (if)j(w)n(e)f(remo)n(v)n(e)e(the)i(lab)r(eled)h(edges)e(corresp)r (onding)523 3425 y(to)38 b(the)h(opp)r(osite)f(assignmen)n(t,)g(then)h (the)g(clause)f(gadget)f(will)i(still)f(con)n(tain)g(a)g(cycle.)523 3525 y(F)-7 b(or)40 b(example,)g(for)g(the)h(assignmen)n(t)e Fi(x)45 b Ft(=)g Fi(y)i Ft(=)d Fi(z)k Ft(=)c(1,)c(w)n(e)g(remo)n(v)n(e) f(edges)h(lab)r(eled)523 3625 y Fi(x)e Ft(=)e(0)p Fi(;)14 b(y)40 b Ft(=)c(0)p Fi(;)14 b(z)40 b Ft(=)d(0.)f(Ho)n(w)n(ev)n(er,)e (the)i(edges)g(lab)r(eled)g Fi(x)i Ft(=)e(1)p Fi(;)14 b(y)40 b Ft(=)c(1)p Fi(;)14 b(z)40 b Ft(=)d(1)f(remain)523 3724 y(and)29 b(form)g(a)g(cycle.)g(So)g(w)n(e)g(m)n(ust)h(remo)n(v)n (e)d(one)i(more)g(of)g(these)g(edges)g(for)g(the)h(resulting)523 3824 y(subgraph)c(of)i(the)g(clause)f(gadget)f(to)i(b)r(e)g(acyclic.) 648 3928 y(\(iii\))21 b(F)-7 b(or)20 b(eac)n(h)g(v)-5 b(ariable)19 b(gadget,)h(if)h(w)n(e)f(remo)n(v)n(e)f(one)h(of)g(the)h (edges)f(in)h(the)g(corresp)r(ond-)523 4028 y(ing)39 b(2-cycle)f(and)h(the)h(corresp)r(onding)d(edge)i(from)g(the)g(clause)g (gadgets)f(represen)n(ting)523 4127 y(clauses)24 b(that)h(con)n(tain)g (this)g(v)-5 b(ariable,)24 b(then)h(the)h(resulting)e(graph)g(do)r(es)h (not)g(con)n(tain)f(an)n(y)523 4227 y(cycle)19 b(comp)r(osed)g(of)g (edges)f(from)h(m)n(ultiple)h(clause)e(gadgets.)g(F)-7 b(or)19 b(the)h(clause)e Fi(x)r Ft(+)r Fi(y)5 b Ft(+)r Fi(z)25 b Ft(=)d(0,)523 4327 y(consider)j(the)i(edge)f(\()p Fi(x)1254 4339 y Fs(2)1292 4327 y Fi(;)14 b(x)1376 4339 y Fs(1)1414 4327 y Ft(\).)27 b(If)g(a)f(cycle)g(con)n(tains)f(this)i (edge,)f(it)h(m)n(ust)g(also)e(con)n(tain)h(the)523 4426 y(only)h(incoming)f(edge)g(to)h(v)n(ertex)f Fi(x)1646 4438 y Fs(2)1684 4426 y Ft(,)h(whic)n(h)f(is)h(the)h(edge)e(lab)r(eled) h Fi(x)c Ft(=)g(1.)k(If)g(these)g(edges)523 4526 y(are)i(con)n(tained)h (in)g(a)g(cycle)g(with)h(edges)e(from)h(another)f(clause)h(gadget,)f (then)i(at)f(v)n(ertex)523 4625 y Fi(x)570 4637 y Fs(1)608 4625 y Ft(,)j(w)n(e)h(can)f(mo)n(v)n(e)f(to)i(another)e(gadget.)h(Ho)n (w)n(ev)n(er,)e(w)n(e)j(arriv)n(e)d(at)j(a)f(v)n(ertex)f(suc)n(h)i (that)523 4725 y(the)c(only)f(out)g(edge)g(corresp)r(onds)e(to)i(the)h (edge)f(that)g(remains)g(i\013)h Fi(x)f Ft(has)g(b)r(een)h(set)f(to)g (0,)523 4825 y(whic)n(h)g(is)f(not)h(the)g(case)e(if)j(the)f(edge)f (lab)r(eled)g Fi(x)e Ft(=)e(1)k(w)n(as)g(presen)n(t.)g(So)g(there)g (cannot)h(b)r(e)523 4924 y(an)n(y)e(cycles)g(that)h(use)f(edges)g(from) g(more)g(than)h(one)f(clause)g(gadget.)p eop %%Page: 15 15 15 14 bop 1487 232 a Fq(On)25 b(Relaxations)h(for)h(the)e(Linear)h (Ordering)g(Problem)198 b(15)648 448 y Ft(It)29 b(follo)n(ws)g(from)g (\(i\),\(ii\),)i(and)e(\(iii\))i(that)e(the)h(minim)n(um)g(feedbac)n(k) f(arc)g(set)g(has)g(size)523 548 y Fi(n)18 b Ft(+)g(3)p Fi(m)g Ft(+)g Fi(u)p Ft(.)2389 b Fh(u)-55 b(t)523 722 y Fg(Corollary)32 b(3.)41 b Fj(The)35 b(Maximum)g(A)l(cyclic)h(Sub)l (gr)l(aph)f(for)h Fi(G)f Fj(is)g(of)h(size)f Fi(n)22 b Ft(+)g(33)p Fi(s)f Ft(+)g(32)p Fi(u)523 822 y Fj(wher)l(e)32 b Fi(s)f Fj(and)h Fi(u)e Fj(r)l(epr)l(esent)h(the)g(numb)l(er)g(of)h (satis\014e)l(d)f(and)h(unsatis\014e)l(d)f(clauses,)h(r)l(esp)l(e)l(c-) 523 922 y(tively,)f(for)g(an)f(assignment)f(that)h(satis\014es)g(the)g (maximum)f(numb)l(er)g(of)i(clauses.)523 1096 y(Pr)l(o)l(of)38 b(of)g(The)l(or)l(em)h(7.)d Ft(By)g(Corollary)d(3)i(and)h(b)n(y)f (Theorem)g(6)g(it)i(is)e(NP-hard)g(to)g(dis-)523 1196 y(tinguish)k(b)r(et)n(w)n(een)f(a)g(graph)g(that)g(has)g(a)h(maxim)n (um)f(acyclic)g(subgraph)f(of)i(size)f Fi(n)25 b Ft(+)523 1295 y(33\()649 1262 y Fs(1)p 648 1276 34 4 v 648 1324 a(2)709 1295 y Ft(+)17 b Fi(\017)p Ft(\))p Fi(m)g Ft(+)g(32\()1155 1262 y Fs(1)p 1154 1276 V 1154 1324 a(2)1214 1295 y Fh(\000)g Fi(\017)p Ft(\))p Fi(m)27 b Ft(and)g(a)g(graph)f(that)h(has)f(a)h (maxim)n(um)g(acyclic)f(subgraph)g(of)523 1395 y(size)f Fi(n)14 b Ft(+)g(33)p Fi(m)p Ft(.)24 b(If)i(w)n(e)f(could)g(appro)n (ximate)f(the)i(maxim)n(um)f(acyclic)g(subgraph)f(to)h(within)533 1462 y Fs(2)p Fd(n)p Fs(+65)p 533 1476 192 4 v 533 1523 a(2)p Fd(n)p Fs(+66)743 1494 y Ft(+)9 b Fi(\017)p Ft(,)21 b(then)i(w)n(e)g(could)f(distinguish)h(b)r(et)n(w)n(een)f(these)h(t)n (w)n(o)f(cases.)g(Therefore)f(it)i(is)f(NP-)523 1594 y(hard)30 b(to)h(appro)n(ximate)e(to)i(appro)n(ximate)e(the)i(maxim)n (um)g(acyclic)f(subgraph)g(to)g(within)533 1661 y Fs(2)p Fd(n)p Fs(+65)p 533 1675 V 533 1722 a(2)p Fd(n)p Fs(+66)751 1694 y Ft(+)17 b Fi(\017)p Ft(.)26 b(W)-7 b(e)27 b(can)g(mak)n(e)f Fi(n)g Ft(arbitrarily)f(small)i(compared)e(to)i Fi(m)g Ft(b)n(y)f(creating)g(another)523 1793 y(set)g(of)h(linear)f(equations) f(in)i(whic)n(h)f(eac)n(h)g(original)f(equation)g(app)r(ears)h Fi(k)j Ft(times)e(for)f(some)523 1893 y Fi(k)k Ft(so)d(that)h(w)n(e)f (ha)n(v)n(e)g Fi(k)s(m)g Ft(clauses)g(and)g(only)g Fi(n)h Ft(v)-5 b(ariables.)26 b(The)h(ratio)2783 1860 y Fs(2)p Fd(n)p Fs(+65)p 2783 1874 V 2783 1922 a(2)p Fd(n)p Fs(+66)3012 1893 y Ft(is)g(arbitrar-)523 2007 y(ily)i(close)f(to)954 1974 y Fs(65)p 954 1988 67 4 v 954 2036 a(66)1059 2007 y Ft(as)g Fi(k)k Ft(b)r(ecomes)c(large.)f(Therefore,)h(it)h(is)g (NP-hard)e(to)i(appro)n(ximate)e(the)523 2107 y(maxim)n(um)g(acyclic)g (subgraph)g(to)g(within)1910 2074 y Fs(65)p 1910 2088 V 1910 2135 a(66)2005 2107 y Ft(+)18 b Fi(\017)27 b Ft(for)h(an)n(y)p Fi(\017)21 b(>)i Ft(0.)735 b Fh(u)-55 b(t)523 2372 y Fk(References)561 2563 y Fq(1.)43 b(Alb)r(erto)33 b(Caprara)h(and)e (Matteo)i(Fisc)n(hetti,)g(1/2-Ch)n(v)l(atal-Gomory)f(cuts,)g Fm(Mathematic)l(al)663 2654 y(Pr)l(o)l(gr)l(amming)p Fq(,)27 b(74A,)f(1996,)h(221{235.)561 2746 y(2.)43 b(P)-6 b(.)34 b(Erd\177)-38 b(os)36 b(and)e(H.)g(Sac)n(hs,)h(Regulare)g (Graphe)g(gegeb)r(ener)g(T)-6 b(aillen)n(w)n(eite)37 b(mit)d(minimaler)663 2837 y(Knotenzahl,)26 b(Wiss.)g(Z.)g(Univ.)f (Halle-Witten)n(b)r(erg,)i Fm(Math.)g(Nat.)f Fq(12,)h(1963,)g(251{258.) 561 2928 y(3.)43 b(M.)34 b(X.)g(Go)r(emans)g(and)f(L.)h(A.)g(Hall,)h (The)f(Strongest)g(F)-6 b(acets)34 b(of)g(the)g(Acyclic)g(Subgraph)663 3020 y(P)n(olytop)r(e)26 b(are)g(Unkno)n(wn,)f Fm(Pr)l(o)l(c)l(e)l(e)l (dings)30 b(of)d(IPCO)g(1996)p Fq(,)f(415{429.)561 3111 y(4.)43 b(M.)36 b(Gr\177)-38 b(o)q(tsc)n(hel,)36 b(M.)g(J)r(\177)-40 b(unger,)37 b(and)e(G.)h(Reinelt,)g(On)f(the)g(Maxim)n(um)f(Acyclic)i (Subgraph)663 3202 y(P)n(olytop)r(e,)26 b Fm(Mathematic)l(al)j(Pr)l(o)l (gr)l(amming)p Fq(,)e(33,)f(1985,)i(28{42.)561 3294 y(5.)43 b(M.)25 b(Gr\177)-38 b(otsc)n(hel,)25 b(M.)g(J)r(\177)-40 b(unger,)25 b(and)f(G.)h(Reinelt,)g(F)-6 b(acets)25 b(of)g(the)f (Linear)h(Ordering)f(P)n(olytop)r(e,)663 3385 y Fm(Mathematic)l(al)k (Pr)l(o)l(gr)l(amming)p Fq(,)f(33,)g(1985,)g(43{60.)561 3476 y(6.)43 b(M.)18 b(Gr\177)-38 b(otsc)n(hel,)20 b(L.)e(Lo)n(v\023) -38 b(asz)19 b(and)e(A.)h(Sc)n(hrijv)n(er,)g Fm(Ge)l(ometric)k(A)n (lgorithms)g(and)f(Combinatorial)663 3568 y(Optimization)p Fq(,)26 b(Springer-V)-6 b(erlag,)26 b(New)g(Y)-6 b(ork,)25 b(1988.)561 3659 y(7.)43 b(Johan)35 b(H)-10 b(\027)-48 b(astad,)36 b(Some)e(Optimal)g(Inappro)n(ximabilit)n(y)f(Results,)j Fm(Pr)l(o)l(c)l(e)l(e)l(dings)i(of)e(STOC)p Fq(,)663 3750 y(1997.)561 3842 y(8.)43 b(Ric)n(hard)36 b(M.)i(Karp,)f (Reducibilit)n(y)g(Among)f(Com)n(binatorial)i(Problems,)g Fm(Complexity)g(of)663 3933 y(Computer)28 b(Computations)p Fq(,)g(Plen)n(um)c(Press,)j(1972.)561 4024 y(9.)43 b(Rudolf)29 b(Mueller,)h(On)e(the)h(P)n(artial)h(Order)f(P)n(olytop)r(e)g(of)h(a)f (Digraph,)g Fm(Mathematic)l(al)j(Pr)l(o-)663 4116 y(gr)l(amming)p Fq(,)26 b(73,)h(1996,)g(31{49.)523 4207 y(10.)43 b(Alan)n(tha)28 b(Newman,)h(Appro)n(ximating)f(the)h(Maxim)n(um)e(Acyclic)i(Subgraph,)g (M.S.)g(Thesis,)663 4298 y(MIT,)d(June)g(2000.)523 4390 y(11.)43 b(S.)32 b(P)n(oljak,)i(P)n(olyhedral)f(and)f(Eigen)n(v)l(alue) g(Appro)n(ximations)g(of)h(the)f(Max-Cut)g(Problem,)663 4481 y Fm(Sets,)c(Gr)l(aphs)h(and)f(Numb)l(ers,)g(Col)t(lo)l(q.)f (Math.)h(So)l(c.)f(Janos)i(Bolyai)k Fq(60,)26 b(1992,)i(569{581.)523 4572 y(12.)43 b(Andreas)31 b(Sc)n(h)n(ulz)h(and)f(Rudolf)h(Mueller,)h (The)f(In)n(terv)l(al)g(Order)f(P)n(olytop)r(e)h(of)h(a)f(Digraph,)663 4664 y Fm(Pr)l(o)l(c)l(e)l(e)l(dings)d(of)f(IPCO)e(1995)p Fq(,)h(50{64.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF