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%!PS-Adobe-2.0 %%Creator: dvips 5.519 Copyright 1986, 1993 Radical Eye Software %%Title: revised.dvi %%CreationDate: Thu Apr 2 16:03:55 1998 %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips revised %DVIPSSource: TeX output 1998.04.02:1603 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N 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b(v)o(eri\014er)h(accepts)g(or)f (rejects)g(based)h(on)f(a)g(simple)i(predicate)g(ev)m(aluated)f(on)75 819 y(these)15 b(bits.)20 b(If)15 b(the)g(input)h(instance)g(is)g(in)f (the)g(NP)g(language,)g(then)g(there)g(is)h(a)e(w)o(a)o(y)g(of)h(enco)q (ding)h(the)75 876 y(PCP)d(witness)i(suc)o(h)f(that)f(regardless)h(of)f (the)h(bits)g(read)f(b)o(y)h(the)g(v)o(eri\014er,)g(the)g(v)o (eri\014er)g(accepts.)20 b(If)14 b(the)75 932 y(input)k(instance)g(is)f (not)f(in)i(the)f(NP)g(language,)g(then)g(an)o(y)g(string)g(giv)o(en)g (as)f(a)h(PCP)f(witness)i(will)g(b)q(e)75 989 y(accepted)c(b)o(y)f(the) h(v)o(eri\014er)g(with)f(probabilit)o(y)i(at)e(most)f(1/2)g (\(probabilit)o(y)j(tak)o(en)e(o)o(v)o(er)f(the)i(coin)g(tosses)75 1045 y(of)k(the)g(v)o(eri\014er\).)29 b(The)18 b(gap)g(b)q(et)o(w)o (een)g(the)g(probabilities)j(that)c(the)h(v)o(eri\014er)h(accepts)f (inputs)h(in)g(the)75 1102 y(NP-language)c(and)g(inputs)g(not)g(in)g (the)g(NP-language)g(is)g(a)f(k)o(ey)g(prop)q(ert)o(y)h(that)f(mak)o 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y(vertex)20 b(c)n(over)f Fp(in)h(b)q(ounded)h (degree)e(graphs,)h(that)e(is,)i(selecting)h(as)e(few)g(as)g(p)q (ossible)i(v)o(ertices)e(in)h(a)75 1610 y(graph)c(of)g(b)q(ounded)j (degree)d(suc)o(h)h(that)f(for)g(eac)o(h)h(edge,)g(at)f(least)h(one)f (of)g(its)h(endp)q(oin)o(ts)h(is)f(selected.)75 1666 y(As)d(v)o(ertex)g(co)o(v)o(er)f(is)i(a)e(sp)q(ecial)j(case)e(of)g(set) g(co)o(v)o(er,)f(this)i(implies)h(that)d(for)g(some)h Fo(\017)f(>)g Fp(0,)h(it)g(is)h(NP-hard)75 1723 y(to)d(appro)o(ximate)h (set)g(co)o(v)o(er)f(within)i(a)f(ratio)f(of)h(1)6 b(+)g Fo(\017)p Fp(.)19 b(Using)13 b(the)g(fact)g(that)f(the)h(graph)g(is)g (of)g(b)q(ounded)75 1779 y(degree,)i(one)g(can)h(also)f(sho)o(w)f(that) h(it)g(is)h(NP-hard)f(to)g(appro)o(ximate)f(max)h Fo(k)q Fp(-co)o(v)o(er)g(within)h(a)f(ratio)f(of)75 1836 y(1)c Fn(\000)g Fo(\017)p Fp(.)146 1892 y(T)l(o)18 b(presen)o(t)h(subsequen)o (t)g(hardness)h(of)e(appro)o(ximation)h(results,)h(w)o(e)e(let)h (TIME\()p Fo(t)p Fp(\))g(denote)g(the)75 1949 y(class)13 b(of)e(languages)i(that)e(ha)o(v)o(e)h(a)g(deterministic)i(algorithm)f (that)e(runs)i(in)g(time)g Fo(t)p Fp(,)f(and)h(let)g(ZTIME\()p Fo(t)p Fp(\))75 2005 y(denote)g(the)g(class)h(of)f(languages)g(that)f (ha)o(v)o(e)h(a)f(probabilistic)k(algorithm)d(that)f(runs)h(in)h(exp)q (ected)h(time)75 2061 y Fo(t)h Fp(\(with)g(zero)f(error\).)20 b(W)l(e)c(shall)g(ignore)g(lo)o(w)g(order)f(terms)g(in)h(the)g(appro)o (ximation)f(ratios)g(presen)o(ted)75 2118 y(b)q(elo)o(w.)146 2174 y(Lund)f(and)g(Y)l(annak)m(akis)h([26)o(])e(sho)o(w)o(ed)g(that)g (set)h(co)o(v)o(er)f(cannot)g(b)q(e)i(appro)o(ximated)e(within)i(a)e (ratio)75 2238 y(of)h(log)8 b Fo(n=)p Fp(4)15 b(unless)g Fo(N)5 b(P)19 b Fn(\032)13 b Fo(T)6 b(I)t(M)f(E)s Fp(\()p Fo(n)739 2221 y Fl(O)q Fm(\()p Fp(p)q(olylog)14 b Fl(n)p Fm(\))975 2238 y Fp(\),)g(and)h(that)f(set)g(co)o(v)o(er)g(cannot)g(b)q (e)h(appro)o(ximated)75 2301 y(within)g(a)e(ratio)g(of)h(log)8 b Fo(n=)p Fp(2)13 b(unless)i Fo(N)5 b(P)19 b Fn(\032)13 b Fo(Z)s(T)6 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(instance)g(and)75 481 y(on)f(random)g(coin)h(tosses)f(of)g(the)h(v)o (eri\014er\),)f(and)h(accepts)f(or)g(rejects)g(based)h(on)g(a)f (predicate)h(ev)m(aluated)75 537 y(on)k(the)g(t)o(w)o(o)e(c)o (haracters.)27 b(More)17 b(generally)l(,)j(one)e(ma)o(y)f(view)h Fo(k)q Fp(-pro)o(v)o(er)g(pro)q(of)f(systems)g(as)h(PCPs)f(in)75 594 y(whic)o(h)f(the)f(PCP)g(witness)g(is)h(partitioned)g(in)o(to)f Fo(k)h Fp(segmen)o(ts,)e(and)i(the)f(v)o(eri\014er)g(reads)g(one)h(c)o (haracter)75 650 y(from)11 b(eac)o(h)h(segmen)o(t.)19 b(In)12 b(the)g(terminology)g(of)g(m)o(ultipro)o(v)o(er)g(pro)q(of)f (systems)h([6)o(],)g(eac)o(h)g(segmen)o(t)f(of)h(the)75 707 y(PCP)j(witness)h(is)g(though)o(t)f(of)g(as)g(b)q(eing)i(con)o (trolled)f(b)o(y)g(one)f(pro)o(v)o(er.)20 b(The)c(con)o(ten)o(ts)f(of)g (the)g(segmen)o(t)75 763 y(are)i(called)i(the)f Fk(str)n(ate)n(gy)f Fp(of)g(the)h(pro)o(v)o(er.)26 b(Reading)19 b(a)e(c)o(haracter)g(from)g (a)h(segmen)o(t)f(corresp)q(onds)h(to)75 819 y(the)c(v)o(eri\014er)h 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y(appro)o(ximation)14 b(results)h(w)o(ere)e(obtain)i(for)e(sev)o (eral)h(problems)h(in)g([18)o(],)f(including)j(a)c(threshold)i(of)f(7)p Fo(=)p Fp(8)75 1215 y(for)k(MAX)f(3SA)l(T.)h(As)g(for)g(set)g(co)o(v)o (er,)g(Raz)g(and)g(Safra)g([31)o(])g(constructed)g(new)g(lo)o(w)g (error)g(constan)o(t)75 1271 y(pro)o(v)o(er)e(pro)q(of)h(systems)g(and) g(used)h(them)f(to)g(sho)o(w)f(that)h(for)f(some)h(constan)o(t)g Fo(c)e(>)i Fp(0,)g(it)g(is)h(NP-hard)75 1328 y(to)h(appro)o(ximate)h (set)g(co)o(v)o(er)f(within)i(a)f(ratio)f(of)h Fo(c)8 b Fp(log)g Fo(n)p Fp(.)34 b(It)20 b(is)h(not)e(kno)o(wn)h(whether)g (hardness)g(of)75 1384 y(appro)o(ximating)15 b(set)g(co)o(v)o(er)g (within)h(a)f(ratio)g(of)f(ln)9 b Fo(n)15 b Fp(\(up)g(to)g(lo)o(w)g (order)g(terms\))f(can)i(b)q(e)f(sho)o(wn)g(under)75 1440 y(the)g(assumption)h(that)e Fo(P)19 b Fn(6)p Fp(=)13 b Fo(N)5 b(P)h Fp(,)15 b(rather)g(than)g Fo(N)5 b(P)18 b Fn(6\032)13 b Fo(T)6 b(I)t(M)f(E)s Fp(\()p Fo(n)1261 1424 y Fl(O)q Fm(\(log)t(log)g Fl(n)p 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y(sho)o(w)16 b(that)f(if)i(the)f (subsets)g(in)h Fn(F)k Fp(are)16 b(c)o(hosen)g(at)g(random,)f(then)i (with)f(high)h(probabilit)o(y)h(ev)o(ery)e(bad)75 2382 y(co)o(v)o(er)d(requires)h(at)f(least)h(roughly)g(log)8 b Fo(m)14 b Fp(subsets,)f(as)h(desired.)20 b(Lund)15 b(and)f(Y)l(annak)m(akis)g(sho)o(w)o(ed)f(ho)o(w)75 2438 y(to)h(reduce)h(t)o(w)o(o)e(pro)o(v)o(er)h(pro)q(of)g(systems)f(for)h (satis\014abilit)o(y)i(of)e(a)g(form)o(ula)g Fo(\036)g Fp(to)g(a)g(collection)i(of)e(sets)g(as)964 2581 y(5)p eop %%Page: 6 6 6 5 bop 75 311 a Fp(describ)q(ed)17 b(ab)q(o)o(v)o(e,)e(suc)o(h)h(that) e(if)i Fo(\036)f Fp(is)h(satis\014able,)g(all)g(sets)f(are)g(co)o(v)o (ered)g(b)o(y)h(the)f(go)q(o)q(d)g(w)o(a)o(y)l(,)g(and)g(if)h Fo(\036)75 368 y Fp(is)g(not)f(satis\014able,)g(most)g(sets)f(need)j (to)d(b)q(e)i(co)o(v)o(ered)f(b)o(y)g(the)g(bad)h(w)o(a)o(y)l(.)146 424 y(T)l(o)k(pro)o(v)o(e)g(a)g(ln)9 b Fo(n)20 b Fp(ratio,)h(w)o(e)g (consider)g(a)f(mo)q(di\014ed)i(construction)f(whic)o(h)h(w)o(e)e(call) 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b(reason)e(wh)o(y)h(w)o(e)75 1158 y(obtain)g(stronger)f(b)q (ounds)h(is)h(that)e(w)o(e)g(in)o(tro)q(duce)i(a)e(new)h(ingredien)o(t) h(in)o(to)f Fo(k)q Fp(-pro)o(v)o(er)f(pro)q(of)g(systems)75 1215 y({)g(that)g(of)g(ha)o(ving)h(t)o(w)o(o)e(di\013eren)o(t)i (acceptance)g(predicates,)h(a)e Fk(str)n(ong)g Fp(acceptance)h (predicate,)g(and)g(a)75 1271 y Fk(we)n(ak)e Fp(acceptance)h (predicate.)22 b(In)16 b(our)f(pro)q(of)g(system,)g(the)h(di\013erence) g(b)q(et)o(w)o(een)g(the)g(case)f(when)h Fo(\036)f Fp(is)75 1328 y(satis\014able)i(and)g(the)f(case)g(when)h Fo(\036)f Fp(is)h(not)f(satis\014able)h(is)g(not)f(only)h(in)g(the)f(acceptance)h (probabilit)o(y)l(,)75 1384 y(but)i(also)f(in)h(the)g(acceptance)g (predicate.)30 b(If)19 b Fo(\036)f Fp(is)h(satis\014able,)h(the)e(pro)o (v)o(ers)g(ha)o(v)o(e)g(a)g(strategy)f(that)75 1440 y(alw)o(a)o(ys)j (satis\014es)h(the)g(strong)f(acceptance)h(predicate)h(\(and)f(hence)h (also)f(the)g(w)o(eak)f(one\).)36 b(If)22 b Fo(\036)e Fp(is)75 1497 y(not)15 b(satis\014able,)i(then)f(an)o(y)g(strategy)e (of)i(the)g(pro)o(v)o(ers)f(satis\014es)h(the)g(w)o(eak)f(acceptance)h (predicate)h(on)75 1553 y(only)j(a)g(small)g(fraction)g(of)f(the)h(p)q (ossible)h(queries)g(of)e(the)h(v)o(eri\014er.)34 b(The)20 b(gap)f(that)g(w)o(e)h(obtain)g(for)75 1610 y(appro)o(ximating)c(set)g (co)o(v)o(er)f(is)i(due)g(in)g(part)e(to)h(the)g(di\013erence)i(in)f (acceptance)f(probabilit)o(y)i(b)q(et)o(w)o(een)75 1666 y(the)k(cases)g(that)f Fo(\036)h Fp(is)g(satis\014able)h(and)f Fo(\036)g Fp(is)h(not)e(satis\014able,)j(and)f(in)f(part)g(to)f(the)h (di\013erence)h(in)75 1723 y(acceptance)16 b(predicate.)146 1779 y(In)e(Section)h(2)e(w)o(e)h(describ)q(e)h(our)f Fo(k)q Fp(-pro)o(v)o(er)f(pro)q(of)h(system.)19 b(It)14 b(is)g(fortunate)f(that)g(w)o(e)h(can)f(in)o(v)o(ok)o(e)h(a)75 1836 y(recen)o(t)i(theorem)f(of)h(Raz)g([30)o(])g(regarding)f (reduction)i(of)f(error)f(b)o(y)h(parallel)h(rep)q(etition.)23 b(In)17 b(con)o(trast,)75 1892 y(Lund)k(and)g(Y)l(annak)m(akis)g(used)g (the)f(more)g(complicated)i(t)o(w)o(o)d(pro)o(v)o(er)g(pro)q(of)h (system)g(of)g(F)l(eige)h(and)75 1949 y(Lo)o(v)m(asz)14 b([12)o(])g(\(the)g(result)g(of)g([30)o(])g(w)o(as)f(not)h(a)o(v)m (ailable)i(at)d(the)h(time\),)g(without)g(actually)h(describing)h(it.) 146 2005 y(In)j(Section)h(3)e(w)o(e)h(explain)i(ho)o(w)d(to)g (construct)h(the)g(partition)g(systems)f(men)o(tioned)i(ab)q(o)o(v)o (e.)30 b(In)75 2061 y(Section)12 b(4)f(w)o(e)g(describ)q(e)i(the)f (reduction)g(from)f(our)g Fo(k)q Fp(-pro)o(v)o(er)f(pro)q(of)h(system)g (to)g(set)g(co)o(v)o(er.)18 b(In)12 b(Section)g(5)75 2118 y(w)o(e)h(sho)o(w)g(that)g Fk(max)i Fo(k)q Fk(-c)n(over)e Fp(has)h(an)f(appro)o(ximation)g(threshold)h(at)f(1)7 b Fn(\000)g Fp(1)p Fo(=e)12 b Fp(under)i(the)g(assumption)75 2174 y(that)h Fo(P)20 b Fn(6)p Fp(=)14 b Fo(N)5 b(P)h Fp(.)21 b(In)16 b(Section)h(6)e(w)o(e)h(analyze)g(the)g(lo)o(w)f(order) h(terms)f(for)g(hardness)h(of)f(appro)o(ximation)75 2231 y(for)i(set)h(co)o(v)o(er.)27 b(Under)19 b(the)f(assumption)g(that)f Fo(N)5 b(P)23 b Fn(6\032)18 b Fo(Z)s(T)6 b(I)t(M)f(E)s Fp(\(2)1295 2214 y Fl(n)1316 2203 y Ff(\021)1336 2231 y Fp(\))17 b(for)g(some)h Fo(\021)g(>)g Fp(0,)g(w)o(e)g(sho)o(w)75 2287 y(that)f(set)h(co)o(v)o(er)g(cannot)f(b)q(e)i(appro)o(ximated)f (within)h(ln)9 b Fo(n)j Fn(\000)g Fo(c)p Fp(\(ln)c(ln)h Fo(n)p Fp(\))1328 2271 y Fm(2)1347 2287 y Fp(,)19 b(for)e(some)h (constan)o(t)f Fo(c)h Fp(that)75 2344 y(dep)q(ends)f(on)e Fo(\021)r Fp(.)964 2581 y(6)p eop %%Page: 7 7 7 6 bop 75 311 a Fq(2)69 b(A)23 b(m)n(ult)o(i)o(-pro)n(v)n(er)e(pro)r (of)k(system)75 413 y Fp(Our)20 b(result)g(is)f(based)h(on)f(a)g (reduction)i(from)d(a)h(m)o(ulti-pro)o(v)o(er)h(pro)q(of)f(system.)32 b(In)20 b(Section)g(2.1)e(w)o(e)75 469 y(describ)q(e)f(the)e(NP-hard)g (problem)h(MAX)f(3SA)l(T-5)h(for)e(whic)o(h)i(w)o(e)f(construct)g(the)g (m)o(ultipro)o(v)o(er)g(pro)q(of)75 526 y(system.)k(This)c(sp)q (eci\014c)h(problem)f(has)g(a)f(regular)g(structure)g(that)g(will)i (later)e(b)q(e)h(used)g(in)g(pro)o(ving)g(the)75 582 y(hardness)e(of)e(appro)o(ximating)i(set)f(co)o(v)o(er.)18 b(In)13 b(Section)g(2.2)f(w)o(e)g(use)g(standard)g(tec)o(hniques)i(to)d (construct)75 639 y(a)16 b(lo)o(w)g(error)g(t)o(w)o(o)f(pro)o(v)o(er)g (pro)q(of)h(system)g(for)f(MAX)h(3SA)l(T-5.)23 b(In)17 b(Section)h(2.3)d(w)o(e)h(construct)g(our)g Fo(k)q Fp(-)75 695 y(pro)o(v)o(er)c(pro)q(of)g(system)h(for)f(MAX)h(3SA)l(T-5.)19 b(This)13 b(pro)q(of)f(system)h(has)f(the)h(new)g(feature)g(of)f(ha)o (ving)h(t)o(w)o(o)75 752 y(di\013eren)o(t)i(acceptance)f(predicates,)h (whic)o(h)h(is)e(used)h(in)h(pro)o(ving)e(the)g(hardness)h(of)f(appro)o (ximating)g(set)75 808 y(co)o(v)o(er.)24 b(W)l(e)17 b(remark)f(that)g (the)h(t)o(w)o(o)e(pro)o(v)o(er)h(pro)q(of)h(system)f(of)h(Section)g (2.2)f(is)h(presen)o(ted)h(only)f(so)f(as)75 864 y(to)f(pro)o(vide)i (in)o(tuition)h(and)e(help)h(in)g(the)f(analysis)h(of)f(the)g(\014nal)h Fo(k)q Fp(-pro)o(v)o(er)e(pro)q(of)h(system,)f(and)i(is)f(not)75 921 y(used)g(elsewhere)g(in)g(our)f(pap)q(er.)75 1043 y Fh(2.1)56 b(The)18 b(underlying)f(NP-complete)f(language)75 1128 y Fp(Our)g(starting)e(p)q(oin)o(t)i(is)g(the)f(problem)h(of)e Fg(MAX)j(3SA)l(T-B)p Fp(.)146 1185 y Fk(Input:)25 b Fp(A)18 b(CNF)g(form)o(ula)g(with)g Fo(n)h Fp(v)m(ariables)g(in)g(whic)o(h)g (ev)o(ery)f(clause)h(con)o(tains)f(at)g(most)f(three)75 1241 y(literals)d(\(a)d(literal)j(is)f(a)f(Bo)q(olean)h(v)m(ariable)h (in)f(either)g(p)q(ositiv)o(e)g(or)f(negated)g(form\),)g(and)g(ev)o (ery)h(v)m(ariable)75 1298 y(app)q(ears)i(in)h(a)f(b)q(ounded)i(n)o(um) o(b)q(er)e(of)g(clauses.)146 1354 y Fk(Output:)h Fp(The)f(maxim)o(um)g (n)o(um)o(b)q(er)h(of)e(clauses)i(that)f(can)g(b)q(e)h(satis\014ed)g (sim)o(ultaneously)g(b)o(y)f(some)75 1411 y(assignmen)o(t)g(to)g(the)g (v)m(ariables.)146 1467 y(The)g(follo)o(wing)h(w)o(ell)g(kno)o(wn)f (theorem)g(app)q(ears)g(in)h([1,)e(28].)75 1573 y Fg(Theorem)h(1)23 b Fk(It)14 b(is)g(MAX-SNP)f(har)n(d)i(to)f(appr)n(oximate)i(MAX)d(3SA)m (T-B:)h(for)g(some)g Fo(\017)g(>)f Fp(0)p Fk(,)h(it)g(is)g(NP-)75 1630 y(har)n(d)19 b(to)h(distinguish)e(b)n(etwe)n(en)f(satis\014able)h (3CNF-B)g(formulas,)i(and)f(3CNF-B)f(formulas)h(in)f(which)75 1686 y(at)f(most)f(an)g Fp(\(1)10 b Fn(\000)g Fo(\017)p Fp(\))p Fk(-fr)n(action)16 b(of)h(the)f(clauses)g(c)n(an)f(b)n(e)h (satis\014e)n(d)f(simultane)n(ously.)146 1793 y Fp(W)l(e)h(w)o(ould)i (lik)o(e)f(to)g(w)o(ork)e(with)i(3CNF-B)g(form)o(ulas)f(that)g(ha)o(v)o (e)h(a)f(v)o(ery)g(regular)h(structure,)g(and)75 1849 y(hence)f(de\014ne)h(the)e(problem)h(of)f Fg(MAX)h(3SA)l(T-5)p Fp(.)146 1906 y Fk(Input:)i Fp(A)13 b(CNF)f(form)o(ula)g(with)g Fo(n)h Fp(v)m(ariables)h(and)e(5)p Fo(n=)p Fp(3)g(clauses,)i(in)f(whic) o(h)g(ev)o(ery)g(clause)g(con)o(tains)75 1962 y(exactly)h(three)g (literals,)h(ev)o(ery)f(v)m(ariable)i(app)q(ears)e(in)h(exactly)f (\014v)o(e)g(clauses,)h(and)f(a)g(v)m(ariable)h(do)q(es)f(not)75 2018 y(app)q(ear)h(in)h(a)f(clause)h(more)f(than)g(once.)146 2075 y Fk(Output:)h Fp(The)f(maxim)o(um)g(n)o(um)o(b)q(er)h(of)e (clauses)i(that)f(can)g(b)q(e)h(satis\014ed)g(sim)o(ultaneously)g(b)o (y)f(some)75 2131 y(assignmen)o(t)g(to)g(the)g(v)m(ariables.)75 2238 y Fg(Prop)q(osition)k(2)j Fk(F)m(or)d(some)h Fo(\017)e(>)h Fp(0)p Fk(,)h(it)f(is)g(NP-har)n(d)h(to)f(distinguish)g(b)n(etwe)n(en)f (satis\014able)g(3CNF-5)75 2294 y(formulas,)h(and)g(3CNF-5)f(formulas)h (in)f(which)h(at)g(most)g(a)f Fp(\(1)12 b Fn(\000)g Fo(\017)p Fp(\))p Fk(-fr)n(action)18 b(of)h(the)g(clauses)f(c)n(an)g(b)n(e)75 2351 y(satis\014e)n(d)d(simultane)n(ously.)964 2581 y Fp(7)p eop %%Page: 8 8 8 7 bop 146 311 a Fg(Pro)q(of:)18 b Fp(The)13 b(pro)q(of)g(uses)g(kno)o (wn)g(tec)o(hniques,)h(and)f(is)g(only)h(sk)o(etc)o(hed)f(b)q(elo)o(w.) 19 b(It)13 b(is)h(based)f(on)g(the)75 368 y(hardness)k(of)g(appro)o (ximating)g(MAX)g(3SA)l(T-B)g(\(Theorem)g(1\).)25 b(W)l(e)17 b(c)o(hange)g(an)g(arbitrary)f(3CNF-B)75 424 y(form)o(ula)f Fo( )h Fp(to)f(a)g(new)g(3CNF-5)f(form)o(ula)h Fo(\036)g Fp(\(on)g(a)g(di\013eren)o(t)h(set)e(of)h(v)m(ariables\).)146 481 y(Consider)g(an)o(y)f(v)m(ariable)i Fo(x)f Fp(and)g(let)g Fo(b)f Fp(b)q(e)h(the)g(n)o(um)o(b)q(er)g(of)f(o)q(ccurrences)i(of)e Fo(x)h Fp(in)g Fo( )r Fp(.)k Fo(b)14 b Fp(is)i(b)q(ounded)75 537 y(from)h(ab)q(o)o(v)o(e)h(b)o(y)f(some)h(univ)o(ersal)h(constan)o (t,)e(and)h(w.l.o.g,)f(w)o(e)g(also)h(assume)g(that)f Fo(b)g Fn(\025)g Fp(2.)27 b(Replace)75 594 y(eac)o(h)21 b(o)q(ccurrence)g(of)f Fo(x)h Fp(b)o(y)f(a)g(fresh)h(v)m(ariable)h Fo(x)947 601 y Fl(i)961 594 y Fp(,)f(for)f(0)h Fn(\024)h Fo(i)f Fn(\024)h Fo(b)13 b Fn(\000)h Fp(1,)21 b(and)g(add)g(the)f(2)p Fo(b)g Fp(clauses)75 650 y(\()p Fo(x)119 657 y Fl(i)143 650 y Fn(_)14 b Fp(\026)-26 b Fo(x)210 657 y Fl(i)p Fm(+1)269 650 y Fp(\),)15 b(\()s(\026)-26 b Fo(x)359 657 y Fl(i)384 650 y Fn(_)10 b Fo(x)450 657 y Fl(i)p Fm(+1)510 650 y Fp(\),)15 b(where)h Fo(i)10 b Fp(+)h(1)k(is)h(computed)h(mo)q(d)f 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b(and)g(\()p Fo(y)d Fn(_)g Fp(\026)-26 b Fo(z)1060 883 y Fm(1)1089 876 y Fn(_)11 b Fp(\026)-26 b Fo(z)1148 883 y Fm(2)1168 876 y Fp(\).)20 b(These)14 b(clauses)i(are)e(satis\014ed)h(only)f(if)75 932 y Fo(y)h Fp(=)e(1,)g(in)i(whic)o(h)g(case)i(\026)-26 b Fo(y)16 b Fp(has)e(no)g(in\015uence)i(on)e(the)g(original)i(clause)f (to)e(whic)o(h)i(it)f(w)o(as)g(added.)20 b(Add)14 b(a)75 989 y(constan)o(t)e(n)o(um)o(b)q(er)h(of)g(additional)h(dumm)o(y)f(v)m (ariables)i Fo(w)1063 996 y Fl(i)1089 989 y Fp(so)e(that)f(the)h(total) g(n)o(um)o(b)q(er)g(of)f(v)m(ariables)j(\(of)75 1045 y(the)h(t)o(yp)q(es)g Fo(x)301 1052 y Fl(i)315 1045 y Fp(,)g Fo(y)366 1052 y Fl(i)380 1045 y Fp(,)g Fo(z)430 1052 y Fl(i)444 1045 y Fp(,)g(and)g Fo(w)595 1052 y Fl(i)609 1045 y Fp(\))g(is)g(divisible)j(b)o(y)d(3,)g(and)g(add)g(dumm)o(y)g (3-CNF)g(clauses)g(that)g(con)o(tain)75 1102 y(distinct)22 b(dumm)o(y)e(v)m(ariables)i Fo(z)629 1109 y Fl(i)664 1102 y Fp(and)f Fo(w)791 1109 y Fl(i)825 1102 y Fp(in)g(p)q(ositiv)o(e) h(form,)f(un)o(til)g(eac)o(h)g(dumm)o(y)g(v)m(ariable)h(o)q(ccurs)75 1158 y(exactly)16 b(\014v)o(e)f(times.)146 1215 y(The)g(ab)q(o)o(v)o(e) g(reduction)h(has)f(the)g(follo)o(wing)h(prop)q(erties)g(\(pro)q(of)f (left)g(to)g(the)g(reader\):)131 1321 y(1.)22 b(The)15 b(reduction)h(tak)o(es)f(p)q(olynomial)h(time.)131 1415 y(2.)22 b(If)15 b Fo( )h Fp(is)g(satis\014able)g(then)g(so)f(is)g Fo(\036)p Fp(.)131 1509 y(3.)22 b(The)15 b(n)o(um)o(b)q(er)g(of)f (clauses)i(increases)g(b)o(y)f(at)f(most)g(a)g(constan)o(t)g(m)o (ultiplicativ)o(e)k(factor.)g(\(This)d(is)189 1565 y(a)g(consequence)h (of)f(the)g(fact)g(that)f(eac)o(h)i(clause)g(in)g Fo( )g Fp(is)g(of)f(b)q(ounded)h(length\).)131 1659 y(4.)22 b(The)d(n)o(um)o(b)q(er)g(of)g(unsatis\014able)h(clauses)g(decreases)g (b)o(y)f(at)f(most)h(a)f(constan)o(t)h(m)o(ultiplicativ)o(e)189 1715 y(factor.)40 b(\(This)23 b(is)g(a)f(consequence)i(of)e(the)h(fact) f(that)f(eac)o(h)i(v)m(ariable)h(app)q(ears)e(a)h(b)q(ounded)189 1772 y(n)o(um)o(b)q(er)15 b(of)g(times)g(in)h Fo( )r Fp(\).)146 1878 y(Prop)q(erties)e(3)f(and)h(4)f(ab)q(o)o(v)o(e)h(imply) h(that)e(if)h(a)f Fo(\016)r Fp(-fraction)h(of)f(the)h(causes)g(of)f Fo( )i Fp(are)e(not)g(satis\014able,)75 1934 y(then)21 b(an)g Fo(\017)p Fp(-fraction)g(of)f(the)h(causes)g(of)f Fo( )i Fp(are)f(not)f(satis\014able,)j(for)d(some)g Fo(\017)h Fp(that)f(dep)q(ends)j(on)d Fo(\016)r Fp(.)75 1991 y(Hence)i(if)g(one)g (could)h(distinguish)g(in)g(p)q(olynomial)g(time)f(b)q(et)o(w)o(een)f (satis\014able)i(3CNF-5)d(form)o(ulas)75 2047 y(and)g(3CNF-5)e(form)o (ulas)h(in)h(whic)o(h)h(at)d(most)h(a)g(\(1)12 b Fn(\000)i Fo(\017)p Fp(\)-fraction)19 b(of)g(the)h(clauses)g(can)f(b)q(e)h (satis\014ed)75 2104 y(sim)o(ultaneously)l(,)c(then)g(one)f(could)i (distinguish)g(in)f(p)q(olynomial)h(time)f(b)q(et)o(w)o(een)f (satis\014able)h(3CNF-B)75 2160 y(form)o(ulas)h(and)i(3CNF-B)e(form)o (ulas)g(in)i(whic)o(h)g(at)e(most)g(a)h(\(1)11 b Fn(\000)i Fo(\016)r Fp(\)-fraction)k(of)h(the)g(clauses)g(can)g(b)q(e)75 2217 y(satis\014ed)e(sim)o(ultaneously)l(.)59 b Fe(2)146 2273 y Fp(F)l(ollo)o(wing)18 b(the)g(NP-hardness)h(result)f(of)g(Prop)q (osition)g(2,)g(w)o(e)g(shall)h(assume)f(that)f(the)i(input)g(to)75 2330 y(the)e(m)o(ultipro)o(v)o(er)g(pro)q(of)f(systems)h(that)f(w)o(e)g (construct)h(is)g(either)h(a)e(satis\014able)i(3CNF-5)e(form)o(ula)h (\(a)75 2386 y Fk(true)e Fp(input\),)f(or)g(a)g(3CNF-5)f(form)o(ula)h (in)h(whic)o(h)g(ev)o(ery)f(assignmen)o(t)g(to)f(the)h(v)m(ariables)i (fails)f(to)e(satisfy)75 2443 y(an)i Fo(\017)p Fp(-fraction)h(of)f(the) g(clauses,)g(for)g(some)g(univ)o(ersal)h(constan)o(t)f Fo(\017)e(>)g Fp(0)i(\(a)f Fk(false)h Fp(input\).)964 2581 y(8)p eop %%Page: 9 9 9 8 bop 75 311 a Fh(2.2)56 b(A)19 b(t)n(w)n(o)g(pro)n(v)n(er)g(pro)r (of)g(system)d(for)j(MAX)g(3SA)-5 b(T-5)75 397 y Fp(Using)14 b(kno)o(wn)g(tec)o(hniques,)h(w)o(e)f(construct)f(a)h(one-round)g(t)o (w)o(o-pro)o(v)o(er)e(pro)q(of)h(system)h(for)f(3SA)l(T-5.)19 b(In)75 454 y(our)c(t)o(w)o(o-pro)o(v)o(er)e(pro)q(of)i(system,)g(the)g (\014rst)g(pro)o(v)o(er)f(receiv)o(es)i(as)f(a)g(query)g(the)h(index)g (of)f(a)g(clause,)h(and)75 510 y(returns)g(as)g(an)h(answ)o(er)f(a)g (sequence)h(of)f(three)h(bits)g(\(i.e.,)f(a)g(v)m(alue)i(b)q(et)o(w)o (een)e(0)g(and)h(7\).)23 b(These)17 b(three)75 567 y(bits)e(can)f(b)q (e)i(view)o(ed)f(as)f(Bo)q(olean)h(assignmen)o(ts)f(to)g(the)g(three)h (v)m(ariables)h(of)e(the)g(clause.)21 b(The)14 b(second)75 623 y(pro)o(v)o(er)i(receiv)o(es)h(as)f(a)g(query)h(the)f(index)i(of)e (a)g(v)m(ariable,)i(and)e(returns)h(one)f(bit)h(as)f(an)g(answ)o(er.)23 b(This)75 679 y(bit)18 b(can)g(b)q(e)g(view)o(ed)h(as)e(a)h(Bo)q(olean) g(assignmen)o(t)g(to)f(the)h(v)m(ariable.)28 b(The)18 b(v)o(eri\014cation)h(pro)q(cedure)f(is)75 736 y(as)d(follo)o(ws.)22 b(The)17 b(v)o(eri\014er)f(selects)h(an)e(index)j(of)d(a)h(clause)g(at) g(random,)f(sends)h(it)g(to)g(the)g(\014rst)f(pro)o(v)o(er,)75 792 y(and)h(selects)g(a)f(random)g(v)m(ariable)i(in)f(the)f(clause,)h (and)g(sends)g(its)f(index)i(to)e(the)g(second)h(pro)o(v)o(er.)k(The)75 849 y(v)o(eri\014er)14 b(in)o(terprets)g(the)g(reply)h(of)e(the)h (\014rst)f(pro)o(v)o(er)g(as)g(an)h(assignmen)o(t)g(to)f(the)g(three)h (v)m(ariables)h(in)g(the)75 905 y(clause,)k(and)e(the)h(reply)g(of)f (the)h(second)g(pro)o(v)o(er)f(as)g(an)g(assignmen)o(t)h(to)e(the)i(v)m (ariable)h(selected)g(from)75 962 y(the)c(clause.)21 b(The)15 b(v)o(eri\014er)h(accepts)g(if)f(the)h(follo)o(wing)f(t)o(w)o (o)f(conditions)j(hold:)131 1068 y(1.)22 b Fg(Clause)17 b(c)o(hec)o(k:)i Fp(the)d(assignmen)o(t)f(sen)o(t)g(b)o(y)g(the)g (\014rst)g(pro)o(v)o(er)f(satis\014es)i(the)f(clause.)131 1162 y(2.)22 b Fg(Consistency)g(c)o(hec)o(k:)27 b Fp(the)19 b(assignmen)o(t)g(sen)o(t)g(b)o(y)h(the)f(second)h(pro)o(v)o(er)e(is)i (iden)o(tical)h(to)e(the)189 1218 y(assignmen)o(t)c(for)f(the)i(same)e (v)m(ariable)j(sen)o(t)e(b)o(y)g(the)g(\014rst)g(pro)o(v)o(er.)75 1324 y Fg(Prop)q(osition)k(3)j Fk(L)n(et)16 b Fo(\036)h Fk(b)n(e)f(a)h(3CNF-5)g(formula)g(and)g(let)f Fo(\017)h Fk(b)n(e)f(the)h(fr)n(action)g(of)f(unsatis\014e)n(d)g(clauses)75 1381 y(in)g(the)i(assignment)d(to)i(the)h(variables)e(that)i (satis\014es)d(the)i(lar)n(gest)g(numb)n(er)f(of)h(clauses.)23 b(Then)16 b(under)75 1437 y(the)f(optimal)g(str)n(ate)n(gy)g(of)g(the)g (pr)n(overs,)g(the)g(veri\014er)f(in)g(the)h(ab)n(ove)g(two)g(pr)n (over)h(pr)n(o)n(of)f(system)f(ac)n(c)n(epts)75 1494 y(with)j(pr)n(ob)n(ability)f Fp(\(1)9 b Fn(\000)i Fo(\017=)p Fp(3\))p Fk(.)146 1600 y Fg(Pro)q(of:)19 b Fp(T)l(o)c(see)g(that)g (regardless)g(of)g(the)g(strategy)f(of)g(the)i(pro)o(v)o(ers)e(the)h (acceptance)h(probabilit)o(y)75 1657 y(is)h(at)f(most)f(\(1)c Fn(\000)g Fo(\017=)p Fp(3\),)16 b(observ)o(e)g(that)g(the)g(strategy)f (of)h(the)h(second)g(pro)o(v)o(er)e(de\014nes)j(an)e(assignmen)o(t)75 1713 y Fo(\037)i Fp(to)f(the)h(v)m(ariables)h(of)e Fo(\036)p Fp(.)27 b(When)18 b(the)g(v)o(eri\014er)g(selects)g(a)f(clause)i(that)e (is)h(not)f(satis\014ed)h(b)o(y)g Fo(\037)g Fp(\(this)75 1769 y(happ)q(ens)d(with)g(probabilit)o(y)g(at)f(least)g Fo(\017)p Fp(\),)g(then)h(in)g(order)f(to)g(pass)g(the)g(clause)h(c)o (hec)o(k,)g(the)f(\014rst)g(pro)o(v)o(er)75 1826 y(m)o(ust)19 b(set)g(at)f(least)i(one)f(of)g(the)h(three)f(v)m(ariables)i (di\013eren)o(tly)f(from)e Fo(\037)p Fp(,)j(and)e(then)h(the)f (consistency)75 1882 y(c)o(hec)o(k)c(fails)h(with)g(probabilit)o(y)g (at)f(least)g(1)p Fo(=)p Fp(3.)146 1939 y(A)f(strategy)g(that)g(guaran) o(tees)g(acceptance)h(probabilit)o(y)h(of)e(at)g(least)h(\(1)8 b Fn(\000)i Fo(\017=)p Fp(3\))k(is)h(to)f(let)i Fo(\037)f Fp(b)q(e)g(an)75 1995 y(assignmen)o(t)i(that)g(satis\014es)h(a)f(\(1)11 b Fn(\000)h Fo(\017)p Fp(\)-fraction)17 b(of)g(the)h(clauses,)g(and)g (to)f(ha)o(v)o(e)g(the)g(\014rst)h(pro)o(v)o(er)e(set)75 2052 y(exactly)g(one)f(v)m(ariable)h(di\013eren)o(tly)h(from)d Fo(\037)i Fp(for)e(clauses)i(not)f(satis\014ed)h(b)o(y)f Fo(\037)p Fp(.)58 b Fe(2)146 2108 y Fp(The)14 b(probabilit)o(y)i(of)e (accepting)i(a)e(false)h(input)g(is)h(kno)o(wn)e(as)g(the)h Fk(err)n(or)g Fp(of)f(the)g(t)o(w)o(o)g(pro)o(v)o(er)f(pro)q(of)75 2165 y(system.)18 b(F)l(or)12 b(the)g(ab)q(o)o(v)o(e)f(t)o(w)o(o-pro)o (v)o(er)f(pro)q(of)h(system,)h(the)g(error)f(ma)o(y)h(b)q(e)g(as)g (high)g(as)g(1)s Fn(\000)s Fo(\017=)p Fp(3.)19 b(W)l(e)12 b(no)o(w)75 2221 y(mo)q(dify)17 b(our)f(construction)h(so)g(as)f(to)g (substan)o(tially)h(lo)o(w)o(er)f(the)h(error.)23 b(This)18 b(is)f(done)g(via)f(a)h(metho)q(d)75 2278 y(kno)o(wn)d(as)g Fk(p)n(ar)n(al)r(lel)h(r)n(ep)n(etition)p Fp(.)k(Rather)14 b(than)h(c)o(ho)q(ose)f(at)g(random)g(one)g(clause,)i(the)e(v)o (eri\014er)h(c)o(ho)q(oses)75 2334 y(at)f(random)f Fo(`)h Fp(clauses)h(\(the)f(v)m(alue)i(of)e Fo(`)g Fp(will)i(b)q(e)e (determined)i(later\).)j(The)c(indices)h(of)e(these)g Fo(`)g Fp(clauses)75 2391 y(are)i(sen)o(t)h(to)f(the)h(\014rst)f(pro)o (v)o(er,)g(who)h(no)o(w)f(replies)j(with)e(a)f(sequence)i(of)f(3)p Fo(`)f Fp(bits.)25 b(F)l(rom)16 b(eac)o(h)h(clause)75 2447 y(the)f(v)o(eri\014er)g(c)o(ho)q(oses)g(at)f(random)h(one)g(v)m (ariable,)h(and)f(sends)g(the)g(indices)i(of)d(the)h Fo(`)g Fp(v)m(ariables)h(to)e(the)964 2581 y(9)p eop %%Page: 10 10 10 9 bop 75 311 a Fp(second)17 b(pro)o(v)o(er.)k(The)16 b(second)h(pro)o(v)o(er)e(replies)j(with)e(a)g(sequence)h(of)e Fo(`)h Fp(bits.)23 b(The)16 b(v)o(eri\014er)h(in)o(terprets)75 368 y(the)c(sequence)h(of)e(bits)h(sen)o(t)f(b)o(y)h(the)f(\014rst)h (pro)o(v)o(er)f(as)g(an)g(assignmen)o(t)h(to)f(the)h(3)p Fo(`)f Fp(v)m(ariables)i(that)e(app)q(ear)75 424 y(in)i(the)g Fo(`)g Fp(random)f(clauses,)h(and)g(in)o(terprets)g(the)g(sequence)h (of)e(bits)h(sen)o(t)f(b)o(y)h(the)g(second)g(pro)o(v)o(er)f(as)g(an)75 481 y(assignmen)o(t)h(to)f(the)h Fo(`)f Fp(v)m(ariables)i(that)f(w)o (ere)f(queried)i(of)e(the)h(second)h(pro)o(v)o(er.)j(The)c(v)o (eri\014er)h(accepts)e(if)75 537 y(the)h(follo)o(wing)h(t)o(w)o(o)e (conditions)j(hold)f(for)e(ev)o(ery)h(one)h(of)f(the)g Fo(`)g Fp(clauses:)20 b(the)15 b(assignmen)o(t)f(sen)o(t)g(b)o(y)g(the) 75 594 y(\014rst)f(pro)o(v)o(er)h(satis\014es)g(the)f(clause,)i(and)f (the)g(assignmen)o(t)g(sen)o(t)f(b)o(y)h(the)g(second)g(pro)o(v)o(er)g (is)g(iden)o(tical)i(to)75 650 y(the)e(assignmen)o(t)h(for)e(the)i (same)f(v)m(ariable)h(sen)o(t)f(b)o(y)h(the)f(\014rst)g(pro)o(v)o(er.) 19 b(Hence)c(from)e(the)i(p)q(oin)o(t)f(of)g(view)75 707 y(of)k(the)i(v)o(eri\014er,)g(the)f(new)g(pro)q(of)f(system)h(is)g (comp)q(osed)g(of)g Fo(`)g Fp(parallel)h(rep)q(etitions)g(of)f(the)g (original)75 763 y(pro)q(of)d(system,)g(where)h(eac)o(h)g(rep)q (etition)h(uses)f(fresh)f(random)h(bits.)24 b(As)17 b(the)f(v)o (eri\014er)i(accepts)e(in)i(the)75 819 y(mo)q(di\014ed)f(pro)q(of)f (system)g(only)g(if)g(all)h(rep)q(etitions)h(are)d(accepting,)i(it)f (is)h(natural)f(to)f(exp)q(ect)i(that)e(the)75 876 y(error)c(of)g(the)g (mo)q(di\014ed)i(pro)q(of)e(system)f(will)j(b)q(e)f(at)f(most)g(\(1)r Fn(\000)r Fo(\017=)p Fp(3\))1210 859 y Fl(`)1226 876 y Fp(.)19 b(Unfortunately)l(,)12 b(this)f(is)h(in)g(general)75 932 y(not)17 b(true,)h(due)g(to)f(subtle)i(reasons)e(that)g(are)g(b)q (est)h(explained)i(b)o(y)d(explicit)j(coun)o(ter)d(examples)i([13)o(].) 75 989 y(Ho)o(w)o(ev)o(er,)e(it)h(is)h(true)f(that)f(parallel)i(rep)q (etition)h(reduces)e(the)g(error)g(at)f(an)h(exp)q(onen)o(tial)h(rate.) 28 b(The)75 1045 y(follo)o(wing)16 b(theorem)f(w)o(as)f(pro)o(v)o(en)h (b)o(y)g(Raz)h([30)o(].)75 1140 y Fg(Theorem)f(4)23 b Fk(If)18 b(a)h(one)g(r)n(ound)g(two)g(pr)n(over)g(pr)n(o)n(of)g(system) f(is)h(r)n(ep)n(e)n(ate)n(d)f Fo(`)h Fk(times)f(indep)n(endently)g(in) 75 1196 y(p)n(ar)n(al)r(lel,)e(then)g(the)g(err)n(or)h(is)f Fp(2)612 1180 y Fy(\000)p Fl(c`)671 1196 y Fk(,)g(wher)n(e)h Fo(c)12 b(>)i Fp(0)i Fk(is)g(a)g(c)n(onstant)g(that)h(dep)n(ends)e (only)h(on)g(the)h(err)n(or)g(of)75 1253 y(the)h(original)e(pr)n(o)n (of)i(system)f(\(assuming)f(this)h(err)n(or)h(was)f(less)f(than)h (one\))g(and)g(on)g(the)h(length)e(of)i(the)75 1309 y(answers)d(of)i (the)f(pr)n(overs)h(in)e(the)i(original)f(pr)n(o)n(of)g(system.)146 1404 y Fp(As)i(the)g(error)f(in)i(our)f(original)h(t)o(w)o(o)e(pro)o(v) o(er)h(pro)q(of)f(system)h(w)o(as)f(a)h(constan)o(t)f(\(1)12 b Fn(\000)g Fo(\017=)p Fp(3\))18 b(that)f(is)75 1460 y(indep)q(enden)o(t)f(of)e Fo(n)p Fp(,)g(and)h(the)f(answ)o(er)f (length)i(w)o(as)e(also)h(a)g(constan)o(t)f(\(three)h(for)g(the)g (\014rst)g(pro)o(v)o(er,)f(one)75 1517 y(for)h(the)g(second)h(pro)o(v)o (er\),)e(it)h(follo)o(ws)g(from)g(Theorem)g(4)g(that)g(the)g(error)f (of)h(our)g(mo)q(di\014ed)i(t)o(w)o(o)d(pro)o(v)o(er)75 1573 y(pro)q(of)i(system)g(is)g(at)g(most)f(2)580 1557 y Fy(\000)p Fl(c`)639 1573 y Fp(,)h(for)f(some)h(univ)o(ersal)i (constan)o(t)d Fo(c)p Fp(.)75 1693 y Fh(2.3)56 b(The)18 b Fd(k)r Fh(-pro)n(v)n(er)g(pro)r(of)h(system)75 1779 y Fp(W)l(e)e(are)g(no)o(w)g(ready)g(to)f(describ)q(e)j(our)e Fo(k)h Fp(pro)o(v)o(er)e(pro)q(of)h(system)g(for)f(MAX)h(3SA)l(T-5)g (whic)o(h)h(has)f(the)75 1836 y(nonstandard)e(feature)g(of)g(t)o(w)o(o) e(di\013eren)o(t)j(acceptance)g(predicates.)k(F)l(or)15 b(reasons)f(of)h(e\016ciency)i(in)f(the)75 1892 y(construction,)i(w)o (e)f(consider)h(a)f(binary)h(co)q(de)g(that)e(con)o(tains)i Fo(k)g Fp(co)q(de)g(w)o(ords,)e(eac)o(h)i(of)f(length)h Fo(`)f Fp(and)75 1949 y(w)o(eigh)o(t)d Fo(`=)p Fp(2,)g(and)g(Hamming)h (distance)g(at)e(least)i Fo(`=)p Fp(3)e(b)q(et)o(w)o(een)i(an)o(y)f(t)o (w)o(o)f(co)q(de)i(w)o(ords.)k(F)l(or)13 b(our)i(main)75 2005 y(result)k(w)o(e)g(shall)h(c)o(ho)q(ose)f Fo(`)g Fp(=)g(\002\(log)8 b(log)h Fo(n)p Fp(\))19 b(and)g Fo(k)h Fp(an)f(arbitrarily)g(large)g(constan)o(t.)30 b(In)20 b(this)f(case,)75 2061 y(assuming)f(w.l.o.g.)26 b(that)17 b Fo(`)h Fp(is)g(an)f(exact)h(p)q(o)o(w)o(er)f(of)g(2)h(and)g(that)f Fo(k)g(<)h(`)p Fp(,)f(the)h(ro)o(ws)f(of)g(a)h(Hadamard)75 2118 y(matrix)e(giv)o(e)h(a)f(co)q(de)i(with)e(the)h(desired)h(prop)q (erties)f(\(in)g(fact,)f(with)h(Hamming)g(distance)g Fo(`=)p Fp(2\).)24 b(F)l(or)75 2174 y(re\014ned)19 b(results)g(\(see)f (Section)i(6\),)e(it)g(is)h(useful)h(to)e(c)o(ho)q(ose)g Fo(k)h(>)f(`)p Fp(,)h(and)g(use)f(some)g(other)g(standard)75 2231 y(co)q(de)e(instead)g(of)e(the)i(Hadamard)e(co)q(de.)146 2287 y(In)h(our)g Fo(k)q Fp(-pro)o(v)o(er)f(pro)q(of)g(system,)h(the)g (v)o(eri\014er)g(selects)h Fo(`)e Fp(clauses)i(uniformly)g(and)f(indep) q(enden)o(tly)75 2344 y(at)k(random.)31 b(Call)21 b(these)e(clauses)h Fo(C)741 2351 y Fm(1)760 2344 y Fo(;)8 b(:)g(:)g(:)d(;)j(C)895 2351 y Fl(`)910 2344 y Fp(.)33 b(F)l(rom)18 b(eac)o(h)h(clause,)i(the)f (v)o(eri\014er)g(selects)g(a)f(single)75 2400 y(v)m(ariable)g (uniformly)f(and)f(indep)q(enden)o(tly)k(at)16 b(random.)26 b(These)18 b(are)f(called)h(the)g Fk(distinguishe)n(d)e Fp(v)m(ari-)75 2457 y(ables)i Fo(x)217 2464 y Fm(1)236 2457 y Fo(;)8 b(:)g(:)g(:)d(;)j(x)364 2464 y 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b(plus)f(three)g(v)m(ariables)h(in)f(eac)o(h)g (of)f(the)h Fo(`=)p Fp(2)f(clauses\).)27 b(F)l(or)17 b(simplicit)o(y)j(in)e(describing)i(the)75 594 y(acceptance)14 b(predicate,)g(w)o(e)f(assume)g(that)g(for)g(eac)o(h)g(of)g(the)g Fo(`=)p Fp(2)g(clauses)h(receiv)o(ed)h(b)o(y)e(the)g(pro)o(v)o(er,)g (the)75 650 y(corresp)q(onding)20 b(bits)f(in)g(the)g(pro)o(v)o(er's)f (answ)o(er)g(enco)q(de)h(a)g(satisfying)g(assignmen)o(t)f(for)g(that)g (clause.)75 707 y(\(This)h(assumption)g(is)h(without)f(loss)g(of)f (generalit)o(y)l(,)i(as)f(whenev)o(er)g(it)g(do)q(es)h(not)e(hold,)i (the)f(v)o(eri\014er)75 763 y(ma)o(y)d(simply)j(complemen)o(t)e(the)g (\014rst)g(of)g(the)g(three)g(bits)h(that)e(corresp)q(ond)h(to)g(the)g (v)m(ariables)h(of)f(the)75 819 y(unsatis\014ed)h(clause,)g(thereb)o(y) e(obtaining)i(a)f(canonical)h(reply)f(that)f(satis\014es)h(the)g (clause.\))25 b(Hence)18 b(in)75 876 y(this)g Fo(k)q Fp(-pro)o(v)o(er)e(pro)q(of)h(system,)g(the)g(acceptance)h(predicates)g 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Fp(.)23 b(No)o(w)15 b(omit)h(the)g(questions)h(on) f(these)g(5)p Fo(`=)p Fp(6)f(co)q(ordinates)i(\(the)f(pro)o(v)o(ers)f (can)75 368 y(reconstruct)j(them)g(an)o(yw)o(a)o(y\).)27 b(It)18 b(follo)o(ws)g(that)g(the)g(t)o(w)o(o)f(pro)o(v)o(ers)g(ha)o(v) o(e)h(a)g(strategy)e(that)i(succeeds)75 424 y(with)h(probabilit)o(y)g (at)f(least)g Fo(\016)r(=k)655 408 y Fm(2)693 424 y Fp(on)g Fo(`=)p Fp(6)f(parallel)j(rep)q(etitions)g(of)d(the)i(original)g(t)o(w) o(o)e(pro)o(v)o(er)g(pro)q(of)75 481 y(system.)i(F)l(rom)c(Theorem)g (4,)f Fo(\016)r(=k)675 464 y Fm(2)708 481 y Fo(<)e Fp(2)778 464 y Fy(\000)p Fl(c`)837 481 y Fp(.)58 b Fe(2)75 624 y Fq(3)69 b(Construction)23 b(of)g(partition)f(systems)75 725 y Fg(De\014nition)d(1)k Fk(A)16 b Fp(partition)g(system)f Fo(B)r Fp(\()p Fo(m;)8 b(L;)g(k)q(;)g(d)p Fp(\))14 b Fk(has)i(the)h(fol)r(lowing)f(pr)n(op)n(erties.)129 830 y(1.)23 b(Ther)n(e)15 b(is)h(a)h(gr)n(ound)f(set)g Fo(B)j Fk(of)d Fo(m)h Fk(p)n(oints.)129 924 y(2.)23 b(Ther)n(e)15 b(is)h(a)h(c)n(ol)r(le)n(ction)e(of)h Fo(L)g Fk(distinct)g Fp(partitions)g Fo(p)1103 931 y Fm(1)1123 924 y Fo(;)8 b(:)g(:)g(:)d(;)j(p)1248 931 y Fl(L)1273 924 y Fk(.)129 1017 y(3.)23 b(F)m(or)14 b Fp(1)e Fn(\024)h Fo(i)f Fn(\024)h Fo(L)p Fk(,)i(p)n(artition)g Fo(p)699 1024 y Fl(i)727 1017 y Fk(is)f(a)h(c)n(ol)r(le)n(ction)e(of)i Fo(k)h Fk(disjoint)e(subsets)f(of)i Fo(B)i Fk(whose)e(union)f(is)g Fo(B)r Fk(.)129 1111 y(4.)23 b(A)o(ny)17 b(c)n(over)h(of)h(the)g Fo(m)f Fk(p)n(oints)g(by)g(subsets)f(that)i(app)n(e)n(ar)g(in)f(p)n (airwise)g(di\013er)n(ent)g(p)n(artitions)g(r)n(e-)189 1167 y(quir)n(es)d(at)i(le)n(ast)e Fo(d)h Fk(subsets.)75 1272 y Fg(Lemm)o(a)e(6)23 b Fk(F)m(or)13 b(every)g Fo(c)f Fn(\025)h Fp(0)g Fk(and)g Fo(m)h Fk(su\016ciently)e(lar)n(ge)h(ther)n (e)g(is)f(a)i Fp(partition)e(system)h Fo(B)r Fp(\()p Fo(m;)8 b(L;)g(k)q(;)g(d)p Fp(\))75 1329 y Fk(whose)16 b(p)n(ar)n(ameters)h(satisfy)e(the)i(fol)r(lowing)f(ine)n(qualities:) 129 1434 y(1.)23 b Fo(L)12 b Fn(')h Fp(\(log)8 b 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y(holds)17 b(since)g Fo(d)d(>>)h(k)q Fp(\).)23 b(Substituting)17 b Fo(L)d Fn(')h Fp(\(log)8 b Fo(m)p Fp(\))1007 2440 y Fl(c)1040 2457 y Fp(and)16 b Fo(d)e(<)h(k)9 b Fp(ln)f Fo(m)p Fp(,)16 b(the)g(probabilit)o(y)i(that)d(some)952 2581 y(12)p eop %%Page: 13 13 13 12 bop 75 313 a Fp(collection)17 b(of)d Fo(d)h Fp(subsets)g(co)o(v)o (ers)f(all)i(p)q(oin)o(ts)f(is)g(at)f(most)g(\(log)8 b Fo(m)p Fp(\))1211 296 y Fl(ck)f Fm(ln)f Fl(m)1320 313 y Fo(e)1341 296 y Fy(\000)p Fl(m)1399 284 y Fc(1)p Ff(=k)1452 313 y Fp(.)20 b(F)l(or)14 b Fo(k)g(<)1686 295 y Fm(ln)5 b Fl(m)p 1657 302 126 2 v 1657 328 a Fm(3)g(ln)g(ln)h Fl(m)1802 313 y Fp(and)75 369 y Fo(m)15 b Fp(su\016cien)o(tly)h(large,) f(this)g(probabilit)o(y)i(tends)e(to)f(0,)h(pro)o(ving)g(that)f(the)h (probabilistic)j(construction)75 425 y(w)o(orks.)146 482 y(The)d(randomized)i(construction)f(describ)q(ed)h(ab)q(o)o(v)o(e)e (requires)h(time)g(p)q(olynomial)h(in)g Fo(m)p Fp(,)e(and)g(suc-)75 538 y(ceeds)21 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y(\(The)h(reader)f(ma)o(y)g(use)h(the)g(follo)o(wing)g(table)g(to)f (translate)g(from)g(our)h(notation)f(to)g(that)g(of)g([27)o(]:)24 b(a)75 1216 y(p)q(oin)o(t)16 b(in)g(set)f Fo(B)j Fn(!)d Fp(a)g(function)h Fo(h)f Fp(in)h(collection)h Fo(H)t Fp(,)e Fo(m)d Fn(!)h(j)p Fo(H)t Fn(j)p Fp(,)h Fo(L)e Fn(!)i Fo(n)p Fp(,)h Fo(k)e Fn(!)g Fo(b)p Fp(,)i Fo(d)d Fn(!)h Fo(k)q Fp(.\))75 1359 y Fq(4)69 b(The)23 b(reduction)f(to)h(set) f(co)n(v)n(er)75 1461 y Fp(Our)16 b(reduction)g(extends)f(that)g(of)g (Lund)h(and)f(Y)l(annak)m(akis)h([26].)146 1517 y(The)g(v)o(eri\014er)h (of)f(the)h Fo(k)q Fp(-pro)o(v)o(er)f(pro)q(of)g(system)g(of)g(Section) h(2.3)e(uses)i(its)g(randomness,)f(whic)o(h)h(w)o(e)75 1573 y(assume)g(that)f(is)i(giv)o(en)f(in)h(form)f(of)f(a)h(random)g (string)g Fo(r)q Fp(,)f(to)h(select)h Fo(`)f Fp(clauses)g(and)h(a)e (distinguished)75 1630 y(v)m(ariable)f(in)f(eac)o(h)f(clause.)20 b(W)l(e)14 b(call)g(these)f Fo(`)h Fp(distinguished)h(v)m(ariables)g (the)e 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y(problem.\))25 b(Eac)o(h)16 b(of)h(the)f Fo(L)h Fp(partitions)g(is)g(lab)q(eled)i(b)o(y)e(an)f Fo(`)p Fp(-bit)i(string)e Fo(p)p Fp(,)h(that)f(corresp)q(onds)h(to)f (an)75 1969 y(assignmen)o(t)d(to)g(the)h(resp)q(ectiv)o(e)g(sequence)h (of)e(distinguished)j(v)m(ariables.)21 b(Eac)o(h)13 b(subset)h(in)g(a)g (partition)75 2025 y(is)20 b(lab)q(eled)i(b)o(y)d(a)g(unique)i(pro)o(v) o(er)e Fo(i)p Fp(.)33 b(W)l(e)19 b(let)h Fo(B)r Fp(\()p Fo(r)o(;)8 b(j;)g(i)p Fp(\))17 b(denote)j(the)g Fo(i)p Fp(th)f(subset)g(of)g(partition)h Fo(j)i Fp(in)75 2082 y(partition)14 b(system)g Fo(r)q Fp(.)19 b(With)14 b(eac)o(h)g (question-answ)o(er)g(pair)h(\()p Fo(q)r(;)8 b(a)p Fp(\))k(of)h(pro)o (v)o(er)h Fo(P)1441 2089 y Fl(i)1455 2082 y Fp(,)g(where)g(1)e Fn(\024)h Fo(i)f Fn(\024)h Fo(k)q Fp(,)h(w)o(e)75 2138 y(asso)q(ciate)g(a)g(subset)g Fo(S)468 2147 y Fm(\()p Fl(q)q(;a;i)p Fm(\))579 2138 y Fp(as)g(follo)o(ws.)19 b(\(Remark:)g(the)c(notation)e Fo(S)1292 2147 y Fm(\()p Fl(q)q(;a;i)p Fm(\))1403 2138 y 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b(of)f(the)952 2581 y(13)p eop %%Page: 14 14 14 13 bop 75 311 a Fp(partitions)18 b(of)f(partition)h(system)g Fo(B)719 318 y Fl(r)738 311 y Fp(\()p Fo(m;)8 b(L;)g(k)q(;)g(d)p Fp(\))14 b(has)k(lab)q(el)h Fo(a)1191 318 y Fl(r)1210 311 y Fp(.)28 b(The)18 b(subset)g Fo(S)1517 321 y Fm(\()p Fl(q)q(;a;i)p Fm(\))1631 311 y Fp(con)o(tains)g(the)75 368 y(p)q(oin)o(ts)e(of)e(subset)i Fo(B)r Fp(\()p Fo(r)o(;)8 b(a)522 375 y Fl(r)540 368 y Fo(;)g(i)p Fp(\),)13 b(for)i(all)h Fo(r)g Fp(with)f(\()p Fo(q)r(;)8 b(i)p Fp(\))j Fn(2)i Fo(r)q Fp(.)146 424 y(Let)21 b Fo(Q)h Fp(denote)g(the)g(n)o(um)o(b)q (er)g(of)f(p)q(ossible)i(di\013eren)o(t)f(questions)g(that)f(a)h(pro)o (v)o(er)e(ma)o(y)h(receiv)o(e.)75 481 y(A)d(question)h(to)e(a)g(single) j(pro)o(v)o(er)d(includes)j Fo(`=)p Fp(2)d(v)m(ariables,)j(for)d(whic)o (h)i(there)f(are)g Fo(n)1576 464 y Fl(`=)p Fm(2)1646 481 y Fp(p)q(ossibilities)75 537 y(\(with)f(rep)q(etition\),)i(and)f Fo(`=)p Fp(2)f(clauses,)h(for)f(whic)o(h)h(there)g(are)f(\(5)p Fo(n=)p Fp(3\))1313 521 y Fl(`=)p Fm(2)1381 537 y Fp(p)q(ossibilities.) 30 b(Hence)18 b Fo(Q)f Fp(=)75 594 y Fo(n)102 577 y Fl(`=)p Fm(2)164 594 y Fn(\001)10 b Fp(\(5)p Fo(n=)p Fp(3\))319 577 y Fl(`=)p Fm(2)369 594 y Fp(.)20 b(Observ)o(e)c(that)e(this)i(n)o (um)o(b)q(er)g(is)f(the)h(same)e(for)h(all)h(pro)o(v)o(ers.)75 674 y Fg(Lemm)o(a)e(7)23 b Fk(If)18 b Fo(\036)g Fk(is)g(satis\014able,) f(then)h(the)g(ab)n(ove)g(set)g(of)g Fo(N)j Fp(=)16 b Fo(mR)i Fk(p)n(oints)f(c)n(an)h(b)n(e)f(c)n(over)n(e)n(d)h(by)g Fo(k)q(Q)75 731 y Fk(subsets.)h(If)13 b(only)h(a)g Fp(\(1)5 b Fn(\000)g Fo(\017)p Fp(\))14 b Fk(fr)n(action)f(of)h(the)h(clauses)e (in)g Fo(\036)h Fk(ar)n(e)g(simultane)n(ously)f(satis\014able,)g(the)h (ab)n(ove)75 787 y(set)i(r)n(e)n(quir)n(es)f Fp(\(1)10 b Fn(\000)g Fp(2)p Fo(f)5 b Fp(\()p Fo(k)q Fp(\)\))p Fo(k)q(Q)j Fp(ln)f Fo(m)16 b Fk(subsets)g(in)f(or)n(der)i(to)g(b)n(e)e (c)n(over)n(e)n(d,)h(wher)n(e)g Fo(f)5 b Fp(\()p Fo(k)q Fp(\))12 b Fn(!)i Fp(0)i Fk(as)g Fo(k)d Fn(!)g(1)p Fk(.)146 867 y Fg(Pro)q(of:)k Fp(If)12 b Fo(\036)f Fp(is)h(satis\014able,)h (consider)f(a)f(satisfying)h(assignmen)o(t)f Fo(A)h Fp(for)f Fo(\036)p Fp(,)g(and)h(\014x)g(for)e(the)i(pro)o(v)o(ers)75 924 y(the)17 b(strategy)e(of)h(answ)o(ering)g(eac)o(h)h(question)g (consisten)o(tly)g(with)g(this)g(satisfying)g(assignmen)o(t.)24 b(No)o(w)75 980 y(consider)14 b(the)f(subsets)h Fo(S)510 990 y Fm(\()p Fl(q)q(;a;i)p Fm(\))606 980 y Fp(,)g(for)e(whic)o(h)i Fo(a)f Fp(is)h(indeed)h(the)e(answ)o(er)g(giv)o(en)g(b)o(y)g(pro)o(v)o (er)g Fo(P)1621 987 y Fl(i)1648 980 y Fp(on)g(question)75 1037 y Fo(q)21 b Fp(under)e(the)g(ab)q(o)o(v)o(e)f(strategy)l(.)30 b(F)l(or)18 b(an)o(y)g Fo(r)q Fp(,)h(consider)h(only)f(the)g(subsets)g Fo(S)1446 1046 y Fm(\()p Fl(q)1476 1051 y Fc(1)1492 1046 y Fl(;a)1521 1051 y Fc(1)1538 1046 y Fl(;)p Fm(1\))1581 1037 y Fp(,)g Fo(S)1641 1046 y Fm(\()p Fl(q)1671 1051 y Fc(2)1688 1046 y Fl(;a)1717 1051 y Fc(2)1734 1046 y Fl(;)p Fm(2\))1777 1037 y Fp(,)g Fo(:)8 b(:)g(:)n Fp(,)75 1093 y Fo(S)103 1103 y Fm(\()p Fl(q)133 1109 y Ff(k)151 1103 y Fl(;a)180 1109 y Ff(k)199 1103 y Fl(;k)q Fm(\))244 1093 y Fp(,)17 b(where)g(for)f(1)g Fn(\024)g Fo(i)f Fn(\024)h Fo(k)q Fp(,)h(\()p Fo(q)743 1100 y Fl(i)757 1093 y Fo(;)8 b(i)p Fp(\))14 b Fn(2)i Fo(r)q Fp(,)h(and)g Fo(a)1038 1100 y Fl(i)1069 1093 y Fp(is)g(the)g(answ)o(er)g(giv)o(en)g(b)o(y)g (pro)o(v)o(er)f Fo(P)1705 1100 y Fl(i)1737 1093 y Fp(on)h(this)75 1150 y(question)i(under)f(the)g(strategy)f(describ)q(ed)j(ab)q(o)o(v)o (e.)28 b(Then)18 b(the)h(partition)f(system)f Fo(B)1591 1157 y Fl(r)1611 1150 y Fp(\()p Fo(m;)8 b(L;)g(k)q(;)g(d)p Fp(\))14 b(is)75 1206 y(completely)g(co)o(v)o(ered)f(b)o(y)g(these)g Fo(k)h Fp(sets)f Fo(S)790 1215 y Fm(\()p Fl(q)820 1220 y Ff(i)832 1215 y Fl(;a)861 1220 y Ff(i)874 1215 y Fl(;i)p Fm(\))911 1206 y Fp(,)g(since)h(for)f(the)g(partition)g(whose)g(lab)q (el)h Fo(p)f Fp(agrees)g(with)75 1263 y(assignmen)o(t)k Fo(A)p Fp(,)h(the)g Fo(i)p Fp(th)f(suc)o(h)h(set)f(con)o(tains)g (subset)h Fo(B)r Fp(\()p Fo(r)o(;)8 b(p;)g(i)p Fp(\),)15 b(for)i(ev)o(ery)g Fo(i)p Fp(.)27 b(A)17 b(similar)i(argumen)o(t)75 1319 y(applies)f(for)d(ev)o(ery)h Fo(r)q Fp(.)23 b(Hence)17 b(the)g(collection)h(of)d(subsets)i(describ)q(ed)h(ab)q(o)o(v)o(e)e(co) o(v)o(ers)f(all)i(p)q(oin)o(ts.)24 b(The)75 1376 y(n)o(um)o(b)q(er)e (of)e(subsets)i(used)g(is)f Fo(k)i Fp(times)e(the)h(n)o(um)o(b)q(er)f (of)g(p)q(ossible)i(questions)f(to)f(a)g(single)h(pro)o(v)o(er.)75 1432 y(In)o(terestingly)l(,)16 b(the)f Fo(k)q(Q)h Fp(subsets)f(used)h (in)g(the)f(co)o(v)o(er)g(happ)q(en)h(to)e(b)q(e)i(disjoin)o(t.)146 1488 y(If)22 b(only)g(a)f(\(1)14 b Fn(\000)h Fo(\017)p Fp(\)-fraction)22 b(of)f(the)h(clauses)g(in)h Fo(\036)f Fp(are)f(sim)o(ultaneously)i(satis\014able,)h(then)e(b)o(y)75 1545 y(Lemma)16 b(5)f(an)o(y)g(strategy)g(of)g(the)h(pro)o(v)o(ers)f (\(w)o(eakly\))g(succeeds)h(with)g(probabilit)o(y)h(at)e(most)g Fo(k)1727 1528 y Fm(2)1757 1545 y Fn(\001)10 b Fp(2)1803 1528 y Fy(\000)p Fl(c`)1862 1545 y Fp(.)75 1601 y(Assume)15 b(a)g(co)o(v)o(er)g(of)g(size)h(\(1)9 b Fn(\000)i Fo(\016)r Fp(\))p Fo(k)q(Q)d Fp(ln)f Fo(m)p Fp(,)15 b(where)g Fo(\016)g Fp(=)e(2)p Fo(f)5 b Fp(\()p Fo(k)q Fp(\),)14 b(and)h(deriv)o(e)h(a)f (con)o(tradiction.)146 1658 y(Let)22 b Fn(C)j Fp(b)q(e)e(a)f (collection)i(of)e(subsets)g(that)g(co)o(v)o(ers)g Fo(S)s Fp(,)h(where)f Fn(jC)s(j)i Fp(=)h(\(1)14 b Fn(\000)h Fo(\016)r Fp(\))p Fo(k)q(Q)8 b Fp(ln)g Fo(m)p Fp(.)41 b(With)75 1714 y(eac)o(h)19 b(question)g Fo(q)i Fp(to)d(a)h(pro)o(v)o (er)f Fo(P)680 1721 y Fl(i)713 1714 y Fp(asso)q(ciate)h(a)f(w)o(eigh)o (t)h Fo(w)1130 1721 y Fl(q)q(;i)1189 1714 y Fp(equal)h(to)e(the)h(n)o (um)o(b)q(er)g(of)g(answ)o(ers)f Fo(a)75 1771 y Fp(suc)o(h)e(that)e Fo(S)304 1780 y Fm(\()p Fl(q)q(;a;i)p Fm(\))414 1771 y Fn(2)f(C)s Fp(.)20 b(Hence)652 1739 y Fb(P)696 1782 y Fl(q)q(;i)744 1771 y Fo(w)777 1778 y Fl(q)q(;i)830 1771 y Fp(=)14 b Fn(jC)s(j)p Fp(.)19 b(With)c(eac)o(h)h(random)f (string)g Fo(r)h Fp(asso)q(ciate)f(a)g(w)o(eigh)o(t)75 1827 y Fo(w)108 1834 y Fl(r)139 1827 y Fp(=)187 1795 y Fb(P)231 1839 y Fm(\()p Fl(q)q(;i)p Fm(\))p Fy(2)p Fl(r)347 1827 y Fo(w)380 1834 y Fl(q)q(;i)421 1827 y Fp(.)j(This)12 b(w)o(eigh)o(t)g(is)g(equal)g(to)f(the)h(n)o(um)o(b)q (er)g(of)f(subsets)g(that)g(participate)i(in)f(co)o(v)o(ering)75 1884 y(the)j Fo(m)g Fp(p)q(oin)o(ts)h(of)f Fo(B)431 1891 y Fl(r)450 1884 y Fp(\()p Fo(m;)8 b(L;)g(k)q(;)g(d)p Fp(\).)17 b(Call)f Fo(r)f Fk(go)n(o)n(d)h Fp(if)f Fo(w)1007 1891 y Fl(r)1038 1884 y Fo(<)e Fp(\(1)d Fn(\000)g Fo(\016)r(=)p Fp(2\))p Fo(k)e Fp(ln)g Fo(m)p Fp(.)75 1964 y Fg(Prop)q(osition)19 b(8)j Fk(The)16 b(fr)n(action)g(of)h(go)n(o)n(d)f Fo(r)h Fk(is)f(at)g(le)n(ast)g Fo(\016)r(=)p Fp(2)p Fk(.)146 2045 y Fg(Pro)q(of:)23 b Fp(Assume)17 b(otherwise.)25 b(Then)827 2012 y Fb(P)871 2056 y Fl(r)897 2045 y Fo(w)930 2052 y Fl(r)964 2045 y Fn(\025)16 b Fp(\(1)11 b Fn(\000)h Fo(\016)r(=)p Fp(2\))1200 2028 y Fm(2)1218 2045 y Fo(k)q(R)c Fp(ln)g Fo(m)16 b(>)g Fp(\(1)11 b Fn(\000)g Fo(\016)r Fp(\))p Fo(k)q(R)c Fp(ln)i Fo(m)p Fp(,)17 b(where)75 2101 y Fo(R)e Fp(denotes)g(the)h(n)o(um)o(b)q(er)f(of)g(p)q(ossible)i (random)e(strings)g(of)g(the)g(v)o(eri\014er.)20 b(On)c(the)f(other)g (hand,)555 2183 y Fb(X)576 2270 y Fl(r)622 2223 y Fo(w)655 2230 y Fl(r)686 2223 y Fp(=)734 2183 y Fb(X)756 2270 y Fl(r)825 2183 y Fb(X)802 2277 y Fm(\()p Fl(q)q(;i)p Fm(\))p Fy(2)p Fl(r)916 2223 y Fo(w)949 2230 y Fl(q)q(;i)1002 2223 y Fp(=)1050 2183 y Fb(X)1061 2274 y Fl(q)q(;i)1123 2193 y Fo(R)p 1123 2213 36 2 v 1123 2255 a(Q)1164 2223 y(w)1197 2230 y Fl(q)q(;i)1250 2223 y Fp(=)1303 2193 y Fo(R)p 1303 2213 V 1303 2255 a(Q)1343 2223 y Fn(jC)s(j)75 2344 y Fp(where)g(the)f(middle)i(equalit)o(y)g(follo)o(ws)e(from)g(the) g(fact)g(that)g(there)g(are)g(exactly)h Fo(R=Q)f Fp(random)g(strings)75 2400 y(that)h(cause)i(the)f(v)o(eri\014er)h(to)e(send)i(out)e(question) i Fo(q)r Fp(.)23 b(Hence)17 b Fn(jC)s(j)c Fo(>)h Fp(\(1)c Fn(\000)h Fo(\016)r Fp(\))p Fo(k)q(Q)d Fp(ln)g Fo(m)p Fp(.)22 b(Con)o(tradiction.)75 2457 y Fe(2)952 2581 y Fp(14)p eop %%Page: 15 15 15 14 bop 75 311 a Fg(Prop)q(osition)19 b(9)j Fk(L)n(et)16 b Fn(C)k Fk(b)n(e)c(a)h(c)n(ol)r(le)n(ction)f(of)h(subsets)e(that)j(c)n (overs)e Fo(S)s Fk(,)g(wher)n(e)h Fn(jC)s(j)c Fp(=)h(\(1)c Fn(\000)g Fo(\016)r Fp(\))p Fo(k)q(Q)e Fp(ln)g Fo(m)p Fk(.)75 368 y(Then)20 b(for)i(some)f(str)n(ate)n(gy)f(for)i(the)f Fo(k)h Fk(pr)n(overs,)g(the)g(veri\014er)f(ac)n(c)n(epts)f Fo(\036)h Fk(with)h(pr)n(ob)n(ability)e(at)i(le)n(ast)75 424 y Fp(2)p Fo(\016)r(=)p Fp(\()p Fo(k)8 b Fp(ln)g Fo(m)p Fp(\))296 408 y Fm(2)315 424 y Fk(.)146 531 y Fg(Pro)q(of:)20 b Fp(Based)c(on)g Fn(C)s Fp(,)f(w)o(e)h(describ)q(e)h(a)e(randomized)i (strategy)e(for)g(the)h Fo(k)g Fp(pro)o(v)o(ers.)21 b(On)16 b(question)75 587 y Fo(q)j Fp(addressed)e(to)f(pro)o(v)o(er)g Fo(P)551 594 y Fl(i)565 587 y Fp(,)g(pro)o(v)o(er)g Fo(P)765 594 y Fl(i)796 587 y Fp(selects)h(an)g(answ)o(er)f Fo(a)h Fp(uniformly)g(at)f(random)h(from)e(the)i(set)75 643 y(of)g(answ)o(ers)f(that)g(satisfy)h Fo(S)571 653 y Fm(\()p Fl(q)q(;a;i)p Fm(\))683 643 y Fn(2)f(C)s Fp(.)25 b(W)l(e)17 b(sho)o(w)f(that)h(under)g(this)h(strategy)d(for)i(the)g(pro)o(v)o (ers,)f(the)75 700 y(v)o(eri\014er)d(w)o(eakly)f(accepts)h(with)f (probabilit)o(y)i(at)d(least)i(2)p Fo(\016)r(=)p Fp(\()p Fo(k)8 b Fp(ln)g Fo(m)p Fp(\))1235 683 y Fm(2)1254 700 y Fp(,)13 b(where)f(this)h(probabilit)o(y)h(is)e(tak)o(en)75 756 y(o)o(v)o(er)k(the)g(join)o(t)h(distribution)h(of)e(the)g(coin)i (tosses)e(of)g(the)g(pro)o(v)o(ers)g(and)h(of)f(the)g(v)o(eri\014er.)25 b(Clearly)l(,)17 b(b)o(y)75 813 y(\014xing)h(the)f(optimal)g(coin)h (tosses)e(for)g(the)h(pro)o(v)o(ers,)f(one)h(obtains)g(a)f (deterministic)j(strategy)d(for)g(the)75 869 y(pro)o(v)o(ers)h(that)g (satis\014es)h(the)g(w)o(eak)f(acceptance)i(predicate)f(with)g(a)g (probabilit)o(y)h(that)e(is)h(at)f(least)h(as)75 926 y(high.)146 982 y(Observ)o(e)e(that)g(for)f(a)h(\014xed)h Fo(r)q Fp(,)f(there)g(is)h(a)f(one)g(to)g(one)g(corresp)q(ondence)h(b)q (et)o(w)o(een)g(sets)f Fo(B)r Fp(\()p Fo(r)o(;)8 b(p;)g(i)p Fp(\))75 1039 y(that)18 b(participate)h(in)g(the)g(co)o(v)o(er)f(of)g Fo(B)758 1046 y Fl(r)777 1039 y Fp(\()p Fo(m;)8 b(L;)g(k)q(;)g(d)p Fp(\))15 b(and)k(sets)f Fo(S)1223 1048 y Fm(\()p Fl(q)q(;a;i)p Fm(\))1338 1039 y Fp(that)g(b)q(elong)i(to)d Fn(C)s Fp(.)30 b(F)l(or)18 b(this)75 1095 y(corresp)q(ondence)f(w)o(e)e(need)i(\()p Fo(q)r(;)8 b(i)p Fp(\))j Fn(2)j Fo(r)j Fp(and)f(the)f(pro)s(jection)h (of)f Fo(a)h Fp(on)f(the)h(sequence)h(of)e(distinguished)75 1152 y(v)m(ariables)i(to)d(b)q(e)i Fo(p)p Fp(.)146 1208 y(Concen)o(trate)g(no)o(w)h(only)g(on)g(go)q(o)q(d)h Fo(r)q Fp(,)f(and)g(compute)g(a)g(lo)o(w)o(er)g(b)q(ound)h(on)f(the)g (probabilit)o(y)i(that)75 1264 y(the)h(v)o(eri\014er)g(accepts)g(when)h (he)f(c)o(ho)q(oses)g(a)f(go)q(o)q(d)h Fo(r)q Fp(.)34 b(Observ)o(e)20 b(that)f(b)o(y)h(prop)q(ert)o(y)g(4)f(of)h(partition)75 1321 y(systems,)14 b(and)g(b)o(y)g(the)h(fact)e(that)h(for)g(go)q(o)q (d)g Fo(r)h Fp(the)f(resp)q(ectiv)o(e)i Fo(B)1206 1328 y Fl(r)1225 1321 y Fp(\()p Fo(m;)8 b(L;)g(k)q(;)g(d)p Fp(\))j(is)k(co)o(v)o(ered)f(b)o(y)g(at)g(most)75 1377 y(\(1)d Fn(\000)i Fo(\016)r(=)p Fp(2\))p Fo(k)7 b Fp(ln)i Fo(m)18 b Fp(subsets,)g(the)g(co)o(v)o(er)f Fn(C)k Fp(m)o(ust)c(ha)o(v) o(e)h(used)g(t)o(w)o(o)f(subsets)h(from)f(the)h(same)g(partition)75 1434 y Fo(p)e Fp(in)h(the)g(co)o(v)o(er)e(of)h Fo(B)453 1441 y Fl(r)472 1434 y Fp(\()p Fo(m;)8 b(L;)g(k)q(;)g(d)p Fp(\))13 b(\(b)o(y)j(our)g(c)o(hoice)h(of)f Fo(\016)g Fp(=)f(2)p Fo(f)5 b Fp(\()p Fo(k)q Fp(\)\).)22 b(Denote)16 b(these)h(t)o(w)o(o)e(subsets)h(b)o(y)75 1490 y Fo(B)r Fp(\()p Fo(r)o(;)8 b(p;)g(i)p Fp(\))14 b(and)j Fo(B)r Fp(\()p Fo(r)o(;)8 b(p;)g(j)s Fp(\))o(,)14 b(where)j Fo(i)d Fn(6)p Fp(=)i Fo(j)s Fp(,)f(and)i(their)g(corresp)q(onding)h (subsets)e(in)i Fn(C)h Fp(b)o(y)d Fo(S)1663 1499 y Fm(\()p Fl(q)1693 1504 y Ff(i)1706 1499 y Fl(;a)1735 1504 y Ff(i)1748 1499 y Fl(;i)p Fm(\))1802 1490 y Fp(and)75 1547 y Fo(S)103 1556 y Fm(\()p Fl(q)133 1561 y Ff(j)149 1556 y Fl(;a)178 1561 y Ff(j)193 1556 y Fl(;j)r Fm(\))235 1547 y Fp(,)i(resp)q(ectiv)o (ely)l(.)27 b(Consider)18 b(what)f(happ)q(ens)h(when)g(the)f(v)o (eri\014er)h(c)o(ho)q(oses)g(random)e(string)i Fo(r)q Fp(.)75 1603 y(Pro)o(v)o(er)c Fo(P)250 1610 y Fl(i)280 1603 y Fp(then)i(receiv)o(es)g(question)g Fo(q)752 1610 y Fl(i)782 1603 y Fp(and)g(pro)o(v)o(er)e Fo(P)1040 1610 y Fl(j)1074 1603 y Fp(receiv)o(es)j(question)f Fo(q)1443 1610 y Fl(j)1461 1603 y Fp(.)21 b(Let)16 b Fo(A)1611 1610 y Fl(r)o(;i)1665 1603 y Fp(denote)g(the)75 1660 y(set)i(of)g(answ)o(ers)g(satisfying)g Fo(a)g Fn(2)h Fo(A)706 1667 y Fl(r)o(;i)763 1660 y Fp(if)g(and)f(only)h(if)g Fo(S)1075 1669 y Fm(\()p Fl(q)1105 1674 y Ff(i)1117 1669 y Fl(;a;i)p Fm(\))1201 1660 y Fn(2)g(C)s Fp(.)28 b(De\014ne)19 b Fo(A)1496 1667 y Fl(r)o(;j)1558 1660 y Fp(in)g(an)f(analogous)75 1716 y(manner.)23 b(By)17 b(the)f(strategy)f(describ)q(ed)j(ab)q(o)o(v) o(e,)e(pro)o(v)o(er)g Fo(P)1105 1723 y Fl(i)1135 1716 y Fp(selects)h(an)f(answ)o(er)g Fo(a)f Fn(2)g Fo(A)1612 1723 y Fl(r)o(;i)1667 1716 y Fp(at)g(random)75 1773 y(\(and)21 b Fo(P)216 1780 y Fl(j)255 1773 y Fp(selects)g Fo(a)h Fn(2)h Fo(A)535 1780 y Fl(r)o(;j)578 1773 y Fp(\).)36 b(Observ)o(e)21 b(that)f(for)g Fo(a)1028 1780 y Fl(i)1063 1773 y Fp(and)h Fo(a)1181 1780 y Fl(j)1220 1773 y Fp(ab)q(o)o(v)o(e,)h Fo(a)1393 1780 y Fl(i)1429 1773 y Fn(2)g Fo(A)1515 1780 y Fl(r)o(;i)1575 1773 y Fp(and)f Fo(a)1693 1780 y Fl(j)1733 1773 y Fn(2)h Fo(A)1819 1780 y Fl(r)o(;j)1862 1773 y Fp(,)75 1829 y(and)17 b(furthermore,)g(for)g(go)q(o)q(d)g Fo(r)q Fp(,)g Fn(j)p Fo(A)716 1836 y Fl(r)o(;i)754 1829 y Fn(j)11 b Fp(+)h Fn(j)p Fo(A)872 1836 y Fl(r)o(;j)915 1829 y Fn(j)j Fo(<)i(k)8 b Fp(ln)h Fo(m)p Fp(.)26 b(Hence)18 b(the)f(join)o(t)g(probabilit)o(y)h(that)f(the)75 1885 y(pro)o(v)o(ers)f(c)o(ho)q(ose)i(to)e(answ)o(er)h(with)g Fo(a)721 1892 y Fl(i)752 1885 y Fp(and)h Fo(a)867 1892 y Fl(j)902 1885 y Fp(is)g(at)f(least)g(4)p Fo(=)p Fp(\()p Fo(k)8 b Fp(ln)g Fo(m)p Fp(\))1315 1869 y Fm(2)1334 1885 y Fp(.)26 b(Since)19 b(b)q(oth)e(these)h(answ)o(ers)75 1942 y(are)d(consisten)o(t)g(with)h(the)f(lab)q(el)i Fo(p)e Fp(of)g(the)g(same)g(partition,)g(the)g(v)o(eri\014er)h(w)o (eakly)f(accepts.)146 1998 y(T)l(o)10 b(complete)i(the)f(pro)q(of,)h (use)f(Prop)q(osition)h(8,)f(whic)o(h)h(sho)o(ws)e(that)h(the)g (probabilit)o(y)h(that)f(a)f(v)o(eri\014er)75 2055 y(c)o(ho)q(oses)15 b(a)g(go)q(o)q(d)g Fo(r)h Fp(is)g(at)e(least)i Fo(\016)r(=)p Fp(2.)57 b Fe(2)146 2111 y Fp(T)l(o)13 b(complete)h(the)g(pro)q(of)g (of)f(Lemma)h(7,)f(observ)o(e)g(that)g(2)p Fo(\016)r(=)p Fp(\()p Fo(k)8 b Fp(ln)g Fo(m)p Fp(\))1334 2095 y Fm(2)1366 2111 y Fo(>)13 b(k)1439 2095 y Fm(2)1466 2111 y Fn(\001)7 b Fp(2)1509 2095 y Fy(\000)p Fl(c`)1568 2111 y Fp(,)14 b(for)f(su\016cien)o(tly)75 2168 y(large)i Fo(`)g Fp(\(made)g(p)q (ossible)i(b)o(y)e Fo(`)e Fp(=)g(\002\(log)8 b(log)g Fo(n)p Fp(\))16 b(and)f Fo(m)d Fp(=)h Fo(n)1134 2151 y Fm(\002\()p Fl(`)p Fm(\))1206 2168 y Fp(\).)58 b Fe(2)146 2224 y Fp(W)l(e)15 b(no)o(w)g(pro)o(v)o(e)f(our)h(main)h(theorem.)75 2330 y Fg(Theorem)f(10)23 b Fk(If)13 b(ther)n(e)g(is)f(some)h Fo(\017)g(>)g Fp(0)g Fk(such)g(that)h(a)g(p)n(olynomial)e(time)i (algorithm)f(c)n(an)g(appr)n(oximate)75 2387 y(set)j(c)n(over)g(within) g Fp(\(1)10 b Fn(\000)g Fo(\017)p Fp(\))e(ln)g Fo(n)p Fk(,)17 b(then)f Fo(N)5 b(P)18 b Fn(\032)13 b Fo(T)6 b(I)t(M)f(E)s Fp(\()p Fo(n)1072 2370 y Fl(O)q Fm(\(log)t(log)g Fl(n)p Fm(\))1253 2387 y Fp(\))p Fk(.)952 2581 y Fp(15)p eop %%Page: 16 16 16 15 bop 146 311 a Fg(Pro)q(of:)17 b Fp(Assume)12 b(that)f(there)g(is) h(a)g(p)q(olynomial)h(time)f(algorithm)f Fo(A)h Fp(that)e(appro)o (ximates)h(set)h(co)o(v)o(er)75 368 y(within)17 b(\(1)9 b Fn(\000)i Fo(\017)p Fp(\))d(ln)g Fo(n)p Fp(.)20 b(Consider)c(no)o(w)f (an)g(arbitrary)g(NP-problem.)21 b(Reduce)16 b(it)g(to)f(the)g (NP-complete)75 424 y(problem)20 b(of)g(appro)o(ximating)f(an)h (instance)h(of)e(max)g(3SA)l(T-5.)34 b(No)o(w)19 b(follo)o(w)h(the)f (reduction)i(to)e(set)75 481 y(co)o(v)o(er)12 b(describ)q(ed)j(ab)q(o)o (v)o(e,)d(with)h Fo(k)h Fp(su\016cien)o(tly)g(large)f(so)g(that)f Fo(f)5 b Fp(\()p Fo(k)q Fp(\))12 b(in)i(Lemma)f(7)f(is)h(smaller)h (than)e Fo(\017=)p Fp(4,)75 537 y(and)i(with)g Fo(m)f Fp(=)g(\(5)p Fo(n)p Fp(\))451 521 y Fm(2)p Fl(`=\017)516 537 y Fp(.)20 b(Using)14 b(the)g(deterministic)i(construction)e(of)f (partition)i(systems)e(describ)q(ed)75 594 y(in)19 b([27)o(],)f(and)g (observing)g(that)f Fo(m)p Fp(,)h Fo(R)g Fp(and)g Fo(Q)g Fp(are)g(b)q(ounded)h(b)o(y)f Fo(n)1260 577 y Fl(O)q Fm(\(log)5 b(log)g Fl(n)p Fm(\))1442 594 y Fp(,)18 b(the)g(time)g(to)f (p)q(erform)75 650 y(this)j(reduction)h(is)f Fo(n)452 634 y Fl(O)q Fm(\(log)5 b(log)h Fl(n)p Fm(\))634 650 y Fp(.)33 b(Recall)22 b(that)d(the)h(n)o(um)o(b)q(er)g(of)f(p)q(oin)o (ts)h(in)h(the)f(set)f(co)o(v)o(er)g(problem)75 707 y(is)h Fo(N)25 b Fp(=)c Fo(mR)e Fp(where)h Fo(R)g Fp(=)h(\(5)p Fo(n)p Fp(\))670 690 y Fl(`)686 707 y Fp(,)f(and)g(observ)o(e)g(that)f (for)g Fo(m)h Fp(as)f(ab)q(o)o(v)o(e,)h(ln)8 b Fo(m)21 b(>)f Fp(\(1)13 b Fn(\000)g Fo(\017=)p Fp(2\))8 b(ln)g Fo(N)d Fp(.)75 763 y(By)17 b(Lemma)g(7,)g(if)h(the)f(original)i(NP)e (instance)h(w)o(as)e(satis\014able,)i(all)g(p)q(oin)o(ts)g(can)f(b)q(e) h(co)o(v)o(ered)f(b)o(y)g Fo(k)q(Q)75 819 y Fp(subsets,)i(and)g(if)g (the)g(original)h(NP)f(instance)g(w)o(as)f(not)g(satis\014able,)j(all)e (p)q(oin)o(ts)g(cannot)g(b)q(e)g(co)o(v)o(ered)75 876 y(b)o(y)i(\(1)13 b Fn(\000)i Fp(2)p Fo(f)5 b Fp(\()p Fo(k)q Fp(\)\))p Fo(k)q(Q)j Fp(ln)f Fo(m)p Fp(.)37 b(F)l(or)21 b(our)f(c)o(hoice)i(of)f Fo(k)h Fp(and)f Fo(m)p Fp(,)h(the)g(ratio)e(b) q(et)o(w)o(een)i(the)f(t)o(w)o(o)e(cases)i(is)75 932 y(\(1)10 b Fn(\000)i Fp(2)p Fo(f)5 b Fp(\()p Fo(k)q Fp(\)\))j(ln)f Fo(m)15 b(>)g Fp(\(1)c Fn(\000)g Fo(\017)p Fp(\))d(ln)g Fo(N)d Fp(.)24 b(Hence)18 b(b)o(y)e(applying)i(algorithm)f Fo(A)f Fp(to)g(the)h(set)g(co)o(v)o(er)f(problem,)75 989 y(one)f(can)h(tell)g(whether)f(the)h(original)g(NP)f(instance)h(w)o (as)e(satis\014able)j(or)d(not.)58 b Fe(2)75 1130 y Fq(5)69 b(Max)24 b Fa(k)r Fq(-co)n(v)n(er)75 1232 y Fp(W)l(e)17 b(sa)o(y)f(that)g(a)g(p)q(olynomial)i(time)f(algorithm)g Fk(c)n(onstructively)g(appr)n(oximates)g Fp(max)f Fo(k)q Fp(-co)o(v)o(er)h(within)75 1288 y(a)e(ratio)f(of)h Fo(\016)f(<)f Fp(1)i(if)g(on)g(an)o(y)g(input,)g(the)g(n)o(um)o(b)q(er)h(of)e(p)q (oin)o(ts)h(co)o(v)o(ered)g(b)o(y)g(the)g Fo(k)h Fp(sets)f(selected)h (b)o(y)f(the)75 1344 y(algorithm)h(is)h(at)f(least)g(a)h Fo(\016)r Fp(-fraction)f(of)g(the)g(n)o(um)o(b)q(er)h(of)f(p)q(oin)o (ts)h(co)o(v)o(ered)f(b)o(y)g(the)h(optimal)g(solution.)75 1401 y(The)e(follo)o(wing)h(prop)q(osition)g(is)g(w)o(ell)g(kno)o(wn)f (and)h(is)f(presen)o(ted)h(for)f(completeness.)75 1493 y Fg(Prop)q(osition)k(11)k Fk(The)17 b(gr)n(e)n(e)n(dy)h(algorithm)g (\(iter)n(atively)f(sele)n(cting)g(the)h(sets)f(that)i(c)n(over)f(the)g (lar)n(gest)75 1550 y(numb)n(er)e(of)h(yet)g(unc)n(over)n(e)n(d)f(p)n (oints\))g(c)n(onstructively)g(appr)n(oximates)h(max)g Fo(k)q Fk(-c)n(over)g(within)g(a)g(r)n(atio)g(of)75 1606 y(at)g(le)n(ast)e Fp(1)10 b Fn(\000)g Fp(1)p Fo(=e)j Fn(')f Fp(0)p Fo(:)p Fp(632)p Fk(.)146 1698 y Fg(Pro)q(of:)17 b Fp(Let)12 b Fo(S)414 1682 y Fy(0)438 1698 y Fn(\032)h Fo(S)h Fp(b)q(e)e(the)g(set)g(of)f(p)q(oin)o(ts)h(co)o(v)o(ered)f(b)o (y)h(the)g(optimal)g(solution,)h(and)e(let)h Fo(n)1722 1682 y Fy(0)1747 1698 y Fp(=)h Fn(j)p Fo(S)1839 1682 y Fy(0)1850 1698 y Fn(j)p Fp(.)75 1755 y(Let)g Fo(n)181 1762 y Fl(i)209 1755 y Fp(b)q(e)h(the)f(n)o(um)o(b)q(er)g(of)g(new)g(p) q(oin)o(ts)h(co)o(v)o(ered)f(b)o(y)g(the)g Fo(i)p Fp(th)g(set)f (selected)j(b)o(y)e(the)g(greedy)g(algorithm.)75 1811 y(Then)j(since)g Fo(S)336 1795 y Fy(0)362 1811 y Fp(can)g(b)q(e)g(co)o (v)o(ered)f(b)o(y)g Fo(k)h Fp(sets,)e(it)i(follo)o(ws)f(that)792 1949 y Fo(n)819 1956 y Fl(i)846 1949 y Fn(\025)899 1913 y Fo(n)926 1896 y Fy(0)948 1913 y Fn(\000)993 1881 y Fb(P)1037 1894 y Fl(i)p Fy(\000)p Fm(1)1037 1926 y Fl(j)r Fm(=1)1108 1913 y Fo(n)1135 1920 y Fl(j)p 899 1939 255 2 v 1013 1980 a Fo(k)146 2046 y Fp(Hence)281 2014 y Fb(P)324 2027 y Fl(i)324 2058 y(j)r Fm(=1)395 2046 y Fo(n)422 2053 y Fl(j)454 2046 y Fn(\025)e Fo(n)529 2030 y Fy(0)551 2046 y Fn(\000)d Fo(n)623 2030 y Fy(0)635 2046 y Fp(\(1)f Fn(\000)737 2029 y Fm(1)p 736 2036 20 2 v 736 2062 a Fl(k)760 2046 y Fp(\))778 2030 y Fl(i)807 2046 y Fp(and)619 2134 y Fl(k)599 2147 y Fb(X)600 2238 y Fl(i)p Fm(=1)666 2187 y Fo(n)693 2194 y Fl(i)720 2187 y Fn(\025)k Fo(n)795 2168 y Fy(0)817 2187 y Fn(\000)e Fo(n)890 2168 y Fy(0)902 2187 y Fp(\(1)e Fn(\000)1004 2156 y Fp(1)p 1002 2177 26 2 v 1002 2218 a Fo(k)1033 2187 y Fp(\))1051 2168 y Fl(k)1084 2187 y Fn(\025)k Fo(n)1159 2168 y Fy(0)1171 2187 y Fp(\(1)d Fn(\000)g Fp(1)p Fo(=e)p Fp(\))183 2287 y Fe(2)146 2344 y Fp(Using)15 b(a)f(T)l(uring)h(reduction,)g(Theorem)g (10)f(can)g(b)q(e)h(used)g(to)f(sho)o(w)g(that)g(the)h(p)q(erformance)f (guar-)75 2400 y(an)o(tee)f(of)f(the)h(greedy)g(algorithm)g(is)g (optimal)g(up)g(to)f(lo)o(w)h(order)g(terms.)18 b(This)c(has)e(also)h (b)q(een)h(observ)o(ed)75 2457 y(b)o(y)h(others)g(\(see)g([15)o(],)g (for)f(example\).)952 2581 y(16)p eop %%Page: 17 17 17 16 bop 75 311 a Fg(Prop)q(osition)19 b(12)k Fk(If)f(max)i Fo(k)q Fk(-c)n(over)f(c)n(an)f(b)n(e)h(c)n(onstructively)f(appr)n (oximate)n(d)i(in)e(p)n(olynomial)h(time)75 368 y(within)16 b(a)h(r)n(atio)f(of)h Fp(\(1)9 b Fn(\000)i Fp(1)p Fo(=e)e Fp(+)i Fo(\017)p Fp(\))16 b Fk(for)h(some)f Fo(\017)d(>)g Fp(0)p Fk(,)j(then)g Fo(N)5 b(P)19 b Fn(\032)13 b Fo(T)6 b(I)t(M)f(E)s Fp(\()p Fo(n)1433 351 y Fl(O)q Fm(\(log)s(log)h Fl(n)p Fm(\))1613 368 y Fp(\))p Fk(.)146 474 y Fg(Pro)q(of:)20 b Fp(Assume)c(that)e(a)i(p)q(olynomial)h(time)f(algorithm)f Fo(A)h Fp(appro)o(ximates)f(max)g Fo(k)q Fp(-co)o(v)o(er)g(within)75 531 y(a)g(ratio)g(of)g(1)10 b Fn(\000)h Fp(1)p Fo(=e)f Fp(+)g Fo(\017)16 b Fp(for)f(some)g Fo(\017)f(>)f Fp(0.)21 b(W)l(e)15 b(use)h(algorithm)g Fo(A)f Fp(as)g(a)g(subroutine)i(in)f(a)f (p)q(olynomial)75 587 y(time)20 b(algorithm)g Fo(B)i Fp(that)d(appro)o(ximates)g(set)h(co)o(v)o(er)f(within)h(\(1)13 b Fn(\000)g Fo(\016)r Fp(\))8 b(ln)g Fo(n)p Fp(.)33 b(By)20 b(Theorem)g(10,)g(this)75 643 y(implies)d(that)e Fo(N)5 b(P)18 b Fn(\032)13 b Fo(T)6 b(I)t(M)f(E)s Fp(\()p Fo(n)653 627 y Fl(O)q Fm(\(log)t(log)g Fl(n)p Fm(\))834 643 y Fp(\).)146 700 y(Giv)o(en)14 b(an)g(instance)h(of)e(set)h(co)o(v)o(er,) f(try)g(out)h(all)h(p)q(ossible)h(v)m(alues)f(of)e(1)g Fn(\024)g Fo(k)g Fn(\024)g Fo(n)h Fp(as)g(the)g(n)o(um)o(b)q(er)g(of)75 756 y(sets)h(that)f(su\016ce)h(to)f(co)o(v)o(er)g(all)i(p)q(oin)o(ts.)k (One)c(of)e(those)h(c)o(hoices)h(of)e Fo(k)i Fp(is)g(the)e(true)h (optimal)h Fo(k)q Fp(,)e(and)h(w)o(e)75 813 y(concen)o(trate)i(on)h (the)f(one)h(case)g(in)g(whic)o(h)h(this)f Fo(k)g Fp(is)h(tried)f(out.) 26 b(Algorithm)18 b Fo(B)j Fp(rep)q(eatedly)d(applies)75 869 y(algorithm)k Fo(A)f Fp(on)h(max)f Fo(k)q Fp(-co)o(v)o(er)g 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g(sets)f(used)i(b)o(y)f(the)g(optim)o(um)75 1208 y(co)o(v)o(er\).)27 b(Simple)19 b(manipulations)h(sho)o(w)d(that)g Fo(`)g Fp(=)g(ln)9 b Fo(n=)p Fp(\(1)i Fn(\000)h Fp(ln)q(\(1)f Fn(\000)i Fo(e\017)p Fp(\)\))k Fo(<)g Fp(\(1)11 b Fn(\000)h Fo(\016)r Fp(\))c(ln)g Fo(n)18 b Fp(for)f(some)75 1264 y Fo(\016)e(>)e Fp(0)h(that)h(dep)q(ends)i(only)e(on)h Fo(\017)p Fp(.)58 b Fe(2)146 1321 y Fp(W)l(e)17 b(sa)o(y)h(that)f(a)g (p)q(olynomial)j(time)e(algorithm)f Fk(appr)n(oximates)i Fp(max)e Fo(k)q Fp(-co)o(v)o(er)h(within)h(a)e(ratio)g(of)75 1377 y(0)12 b Fo(<)h(\016)i(<)e Fp(1)h(if)i(on)f(an)o(y)f(input,)i(the) f(algorithm)g(outputs)f(a)h(n)o(um)o(b)q(er)g(that)f(is)h(b)q(et)o(w)o (een)h(opt)e(and)h Fo(\016)d Fn(\001)d Fp(opt)o(,)75 1434 y(where)j(opt)g(denotes)g(n)o(um)o(b)q(er)h(of)e(p)q(oin)o(ts)i (co)o(v)o(ered)f(b)o(y)g(the)g(optimal)h(solution.)19 b(The)12 b(follo)o(wing)h(theorem)75 1490 y(impro)o(v)o(es)j(on)h(Prop) q(osition)g(12)f(in)h(t)o(w)o(o)e(resp)q(ects:)23 b(appro)o(ximation)16 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(reducing)h(the)e(n)o(um)o(b)q(er)h(of)f(random)g(bits)h(in)g(t)o(w)o (o)e(pro)o(v)o(er)75 1553 y(pro)q(of)f(systems)g(are)f(discussed)j(in)f ([10)o(].)75 1672 y Fh(Ac)n(kno)n(wledgemen)n(ts)75 1758 y Fp(I)i(thank)f(Moni)h(Naor,)f(Leonard)h(Sh)o(ulman,)h(and)f(Ara)o (vind)g(Sriniv)m(asan)h(for)e(a)h(preview)g(of)f([27)o(],)h(and)75 1814 y(Mihir)e(Bellare)h(for)d(his)i(commen)o(ts)f(on)g(an)g(earlier)h (v)o(ersion)g(of)e(this)i(man)o(uscript.)75 1955 y Fq(References)98 2056 y Fp([1])21 b(S.)11 b(Arora,)f(C.)f(Lund,)k(R.)d(Mot)o(w)o(ani,)g (M.)f(Sudan,)j(M.)e(Szegedy)l(.)h(\\Pro)q(of)e(v)o(eri\014cation)i(and) g(hardness)168 2112 y(of)18 b(appro)o(ximation)g(problems".)g(In)h Fk(Pr)n(o)n(c.)f(of)h(33r)n(d)g(A)o(nnual)f(Symp)n(osium)h(on)g(F)m (oundations)f(of)168 2169 y(Computer)f(Scienc)n(e,)e(14{23,)i(1992)p Fp(.)98 2256 y([2])k(S.)14 b(Arora,)e(S.)h(Safra.)f(\\Probabilistic)j (c)o(hec)o(king)g(of)d(pro)q(ofs:)19 b(a)13 b(new)g(c)o (haracterization)h(of)f(NP".)f(In)168 2313 y Fk(Pr)n(o)n(c.)k(of)h(33r) n(d)f(A)o(nnual)g(Symp)n(osium)g(on)g(F)m(oundations)f(of)i(Computer)g (Scienc)n(e,)d(2{13,)k(1992)p Fp(.)98 2400 y([3])j(L.)e(Babai,)g(L.)g (F)l(ortno)o(w,)e(C.)h(Lund,)i(\\Non-deterministic)g(exp)q(onen)o(tial) g(time)f(has)f(t)o(w)o(o-pro)o(v)o(er)168 2457 y(in)o(teractiv)o(e)e (proto)q(cols",)e Fk(Computational)j(Complexity,)f(1:3{40,)i(1991)p Fp(.)952 2581 y(20)p eop %%Page: 21 21 21 20 bop 98 311 a Fp([4])21 b(L.)11 b(Babai,)g(S.)g(Moran.)e(\\Arth)o (ur-Merlin)i(games:)17 b(a)10 b(randomized)h(pro)q(of)f(system,)h(and)g (a)f(hierarc)o(h)o(y)168 368 y(of)15 b(complexit)o(y)h(classes.")f Fk(J.)h(Computer)h(and)g(Sys.)e(Sci.)g Fp(36)f(\(1988\),)f(254{276.)98 462 y([5])21 b(M.)c(Bellare,)i(S.)e(Goldw)o(asser,)g(C.)g(Lund,)i(A.)e (Russell.)i(\\E\016cien)o(t)f(probabilistically)i(c)o(hec)o(k)m(able) 168 518 y(pro)q(ofs)15 b(and)g(applications)i(to)e(appro)o(ximation".)f (In)i Fk(Pr)n(o)n(c.)f(of)i(25th)g(A)o(nnual)e(A)o(CM)g(Symp)n(osium) 168 575 y(on)h(the)h(The)n(ory)f(of)g(Computing)p Fp(,)f(294{304,)e (1993.)98 668 y([6])21 b(M.)16 b(Ben-Or,)h(S.)f(Goldw)o(asser,)g(J.)g (Kilian,)i(A.)e(Wigderson.)g(\\Multi-pro)o(v)o(er)g(in)o(teractiv)o(e)h (pro)q(ofs:)168 725 y(ho)o(w)e(to)f(remo)o(v)o(e)g(in)o(tractabilit)o (y)i(assumptions".)f(In)g Fk(Pr)n(o)n(c.)h(of)g(20th)h(A)o(nnual)e(A)o (CM)g(Symp)n(osium)168 781 y(on)h(the)h(The)n(ory)f(of)g(Computing)p Fp(,)f(113{131,)e(1988.)98 875 y([7])21 b(V.)d(Ch)o(v)m(atal.)h(\\A)f (greedy)g(heuristic)i(for)e(the)g(set)g(co)o(v)o(ering)h(problem".)f Fk(Math.)i(Op)n(er.)f(R)n(es.,)g(4,)168 932 y(1979,)f(233{235.)98 1025 y Fp([8])j(M.)14 b(Dy)o(er,)g(A.)g(F)l(rieze.)h(\\A)f(simple)i (heuristic)g(for)e(the)h Fo(p)p Fp(-cen)o(ter)f(problem".)h Fk(Op)n(er.)g(R)n(es.)g(L)n(ett.,)g(3,)168 1082 y(1985,)j(285{288.)98 1176 y Fp([9])j(U.)d(F)l(eige,)h(S.)e(Goldw)o(asser,)h(L.)g(Lo)o(v)m (asz,)g(S.)g(Safra,)f(S.)h(Szegedy)l(.)h(\\In)o(teractiv)o(e)e(pro)q (ofs)h(and)g(the)168 1232 y(hardness)f(of)g(appro)o(ximating)f (cliques".)i Fk(Journal)g(of)g(the)g(A)o(CM,)e(V)m(ol.)h(43,)h(No.)g (2,)g(1996,)h(268{)168 1289 y(292.)75 1382 y Fp([10])i(U.)14 b(F)l(eige,)g(J.)f(Kilian.)i(\\Imp)q(ossibilit)o(y)h(results)e(for)f (recycling)i(random)e(bits)h(in)g(t)o(w)o(o)f(pro)o(v)o(er)f(pro)q(of) 168 1439 y(systems".)17 b Fk(Pr)n(o)n(c.)g(of)i(27th)f(A)o(nnual)g(A)o (CM)f(Symp)n(osium)h(on)g(the)g(The)n(ory)g(of)h(Computing)p Fp(,)e(457{)168 1495 y(468,)d(1995.)75 1589 y([11])21 b(U.)13 b(F)l(eige,)h(J.)f(Kilian.)i(\\Zero)d(kno)o(wledge)i(and)f(the) g(c)o(hromatic)g(n)o(um)o(b)q(er".)g Fk(Pr)n(o)n(c.)g(of)i(11th)g(A)o (nnual)168 1645 y(IEEE)h(Confer)n(enc)n(e)e(on)i(Computational)h (Complexity)p Fp(,)d(1996.)75 1739 y([12])21 b(U.)h(F)l(eige,)h(L.)f (Lo)o(v)m(asz.)f(\\Tw)o(o)f(pro)o(v)o(er)h(one)h(round)f(pro)q(of)h (systems:)32 b(their)22 b(p)q(o)o(w)o(er)f(and)g(their)168 1796 y(problems".)f(In)g Fk(Pr)n(o)n(c.)f(of)i(24th)g(A)o(nnual)e(A)o (CM)g(Symp)n(osium)h(on)g(the)g(The)n(ory)g(of)g(Computing)p Fp(,)168 1852 y(733{744,)13 b(1992.)75 1946 y([13])21 b(L.)c(F)l(ortno)o(w,)f(J.)g(Romp)q(el,)j(M.)d(Sipser,)h(\\On)h(the)e (P)o(o)o(w)o(er)g(of)g(Multi-Pro)o(v)o(er)h(In)o(teractiv)o(e)g(Proto-) 168 2002 y(cols",)e Fk(The)n(or)n(etic)n(al)g(Computer)i(Scienc)n(e)d Fp(134,)g(545{557,)f(1994.)75 2096 y([14])21 b(S.)16 b(Goldw)o(asser,)e(S.)h(Micali,)i(C.)d(Rac)o(k)o(o\013.)g(\\The)i(kno)o (wledge)f(complexit)o(y)i(of)d(in)o(teractiv)o(e)i(pro)q(of-)168 2153 y(systems.")e Fk(SIAM)h(J.)h(Comp.)g Fg(18)f Fp(\(1989\),)e (186{208.)75 2247 y([15])21 b(S.)c(Guha,)f(S.)h(Kh)o(uller.)h(\\Greedy) e(strik)o(es)h(bac)o(k:)22 b(impro)o(v)o(ed)17 b(facilit)o(y)h(lo)q (cation)f(algorithms".)f(In)168 2303 y Fk(Pr)n(o)n(c.)g(Ninth)g(A)o (nnual)f(A)o(CM-SIAM)f(Symp)n(osium)j(on)f(Discr)n(ete)f(A)o (lgorithms,)h(1998)p Fp(.)75 2397 y([16])21 b(M.)e(Halldorsson.)g (\\Appro)o(ximating)g(the)h(minim)o(um)g(maximal)f(indep)q(endenc)q(e)j (n)o(um)o(b)q(er".)d Fk(In-)168 2453 y(form.)e(Pr)n(o)n(c)n(ess.)e(L)n (ett.)g(46)i(\(1993\))f(169{172)p Fp(.)952 2581 y(21)p eop %%Page: 22 22 22 21 bop 75 311 a Fp([17])21 b(J.)14 b(H)-6 b(\027)-28 b(astad.)13 b(\\Clique)i(is)f(hard)g(to)f(appro)o(ximate)g(within)i Fo(n)1180 295 y Fm(1)p Fy(\000)p Fl(\017)1242 311 y Fp(.")e(In)h Fk(Pr)n(o)n(c.)h(37th)g(IEEE)f(Symp.)h(on)168 368 y(F)m(oundations)h (of)g(Computer)h(Scienc)n(e)p Fp(,)d(627{636,)e(1996.)75 457 y([18])21 b(J.)14 b(H)-6 b(\027)-28 b(astad.)12 b(\\Some)h(optimal) g(inappro)o(ximabilit)o(y)i(results".)e(In)h Fk(pr)n(o)n(c.)g(29th)h(A) o(nnual)f(A)o(CM)f(Sym-)168 513 y(p)n(osium)k(on)f(The)n(ory)g(of)g (Computing)p Fp(,)f(1{10,)f(1997.)75 602 y([19])21 b(J.)c(Hastad,)e(S.) i(Phillips,)i(S.)d(Safra.)f(\\A)h(w)o(ell)i(c)o(haracterized)f(appro)o (ximation)f(problem".)h Fk(Pr)n(o)n(c.)168 658 y(2nd)g(Isr)n(ael)e (Symp.)h(on)g(The)n(ory)g(of)g(Computing)h(and)f(Systems,)f(261{265,)j (1993)p Fp(.)75 747 y([20])j(D.)15 b(Ho)q(c)o(h)o(baum)h(\(editor\).)f (Appro)o(ximation)g(Algorithms)h(for)f(NP-hard)h(Problems.)f(PWS)g (Pub-)168 803 y(lishing)j(Compan)o(y)l(,)c(Boston,)g(1997.)75 892 y([21])21 b(D.)15 b(Ho)q(c)o(h)o(baum,)g(D.)g(Shmo)o(ys.)g(\\A)g(b) q(est)h(p)q(ossible)h(appro)o(ximation)f(algorithm)f(for)g(the)h Fo(k)q Fp(-cen)o(ter)168 948 y(problem".)g Fk(Math.)g(Op)n(er.)g(R)n (es.,)g(10,)g(1985,)i(180{184.)75 1037 y Fp([22])j(W.)16 b(Hsu,)g(G.)g(Nemhauser.)g(\\Easy)f(and)i(hard)f(b)q(ottlenec)o(k)h(lo) q(cation)g(problems".)f Fk(Discr)n(ete)h(Ap-)168 1094 y(plie)n(d)f(Math.,)h(1,)f(1979,)h(209{216.)75 1182 y Fp([23])k(D.)13 b(Johnson.)g(\\Appro)o(ximation)h(algorithms)f(for)f (com)o(binatorial)i(problems".)f Fk(J.)h(Comput.)i(Sys-)168 1239 y(tem)h(Sci.)e(9,)i(1974,)g(256{278)p Fp(.)75 1328 y([24])k(L.)14 b(Lo)o(v)m(asz.)g(\\On)g(the)g(ratio)g(of)f(the)h (optimal)h(in)o(tegral)f(and)g(fractional)g(co)o(v)o(ers".)f Fk(Discr)n(ete)h(Math-)168 1384 y(ematics)j(13)f(\(1975\))h(383{390)p Fp(.)75 1473 y([25])k(C.)16 b(Lund,)i(L.)f(F)l(ortno)o(w,)e(H.)h (Karlo\013,)g(N.)g(Nisan.)h(\\Algebraic)h(Metho)q(ds)e(for)g(In)o (teractiv)o(e)h(Pro)q(of)168 1529 y(Systems.")e Fk(J.)h(A)o(CM)p Fp(,)e Fg(39)h Fp(\(1992\),)e(859{868.)75 1618 y([26])21 b(C.)13 b(Lund,)i(M.)d(Y)l(annak)m(akis.)i(\\On)g(the)g(hardness)f(of)g (appro)o(ximating)h(minimization)h(problems".)168 1674 y Fk(JA)o(CM)g(41\(5\),)i(1994,)g(960{981.)75 1763 y Fp([27])k(M.)c(Naor,)f(L.)g(Sc)o(h)o(ulman,)i(A.)f(Sriniv)m(asan.)h (\\Splitters)f(and)g(near-optimal)h(derandomization".)168 1819 y Fk(Pr)n(o)n(c.)d(of)h(36th)g(A)o(nnual)f(Symp)n(osium)g(of)h(F)m (oundations)f(of)h(Computer)g(Scienc)n(e)p Fp(,)c(182{191,)g(1995.)75 1908 y([28])21 b(C.)k(P)o(apadimitriou,)j(M.)d(Y)l(annak)m(akis.)h (\\Optimization,)i(appro)o(ximation,)g(and)d(complexit)o(y)168 1965 y(classes".)15 b Fk(JCSS,)g(43,)i(1991,)g(425{440.)75 2053 y Fp([29])k(A.)15 b(P)o(az,)g(S.)g(Moran.)f(\\Nondeterministic)j (p)q(olynomial)g(optimization)f(problems)g(and)f(their)h(ap-)168 2110 y(pro)o(ximations",)f Fk(The)n(or)n(etic)n(al)g(Computer)i(Scienc) n(e,)e(15)h(\(1981\))h(251{277)p Fp(.)75 2199 y([30])k(R.)e(Raz.)g(\\A) g(parallel)i(rep)q(etition)f(theorem".)e Fk(Pr)n(o)n(c.)h(of)h(27th)g (A)o(nnual)f(A)o(CM)f(Symp)n(osium)i(on)168 2255 y(the)d(The)n(ory)f (of)g(Computing)p Fp(,)f(447{456,)e(1995.)75 2344 y([31])21 b(R.)c(Raz,)f(S.)g(Safra.)f(\\A)h(sub-constan)o(t)g(error-probabilit)o (y)h(lo)o(w-degree)f(test,)g(and)g(sub-constan)o(t)168 2400 y(error-probabilit)o(y)e(PCP)e(c)o(haracterization)h(of)f(NP".)g Fk(Pr)n(o)n(c.)h(of)h(29th)g(A)o(nnual)f(A)o(CM)g(Symp)n(osium)168 2457 y(on)j(The)n(ory)h(of)f(Computing)p Fp(,)f(475{484,)e(1997.)952 2581 y(22)p eop %%Page: 23 23 23 22 bop 75 311 a Fp([32])21 b(A.)15 b(Shamir.)h(\\IP=PSP)l(A)o(CE".)e Fk(Journal)j(of)f(the)h(A)o(CM,)e(39:869{877,)j(1992)p Fp(.)75 405 y([33])j(P)l(.)13 b(Sla)o(vik.)f(\\A)h(tigh)o(t)f(analysis) h(of)f(the)g(greedy)g(algorithm)h(for)f(set)g(co)o(v)o(er".)f Fk(Pr)n(o)n(c.)i(of)h(28th)g(A)o(nnual)168 462 y(A)o(CM)i(Symp)n(osium) g(on)g(The)n(ory)g(of)g(Computing)p Fp(,)f(435{439,)e(1996.)75 555 y([34])21 b(A.)15 b(Sriniv)m(asan.)h(\\Impro)o(v)o(ed)e(appro)o (ximations)h(of)f(pac)o(king)h(and)g(co)o(v)o(ering)g(problems".)g Fk(Pr)n(o)n(c.)g(of)168 612 y(27th)i(A)o(nnual)f(A)o(CM)f(Symp)n(osium) h(on)g(the)h(The)n(ory)f(of)g(Computing)p Fp(,)f(268{276,)e(1995.)952 2581 y(23)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF