(original) (raw)

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y(the)f(usual)f(reduction)g(from)h(MAX)h (3SA)-8 b(T)35 b(to)g(Indep)s(enden)m(t)d(Set\(IS\))j(is)e(gap)i (preserving.)50 b(Let)35 b Fm(\036)g Fp(b)s(e)0 3454 y(an)40 b(instance)g(of)h(MAX3SA)-8 b(T)41 b(with)e Fm(n)h Fp(v)-5 b(ariables)39 b(and)h Fm(m)g Fp(clauses.)71 b(W)-8 b(e)41 b(construct)g(the)g(graph)e Fm(G)3553 3469 y Fh(\036)0 3567 y Fp(from)28 b Fm(\036)i Fp(as)f(follo)m(ws.)39 b(The)29 b(graph)f Fm(G)1253 3582 y Fh(\036)1328 3567 y Fp(has)h(a)g(v)m(ertex)i Fm(v)1882 3581 y Fh(i;j)1991 3567 y Fp(for)d(ev)m(ery)i(o)s(ccurrence)f(of)g(the)g(v)-5 b(ariable)28 b Fm(x)3467 3581 y Fh(i)3524 3567 y Fp(in)0 3680 y(clause)k Fm(C)335 3694 y Fh(j)371 3680 y Fp(.)45 b(All)30 b(the)i(v)m(ertices)h(corresp)s(onding)d(to)i(literals)e(from) i(the)g(same)g(clause)g(are)g(joined)e(b)m(y)i(an)0 3793 y(edge)f(\(th)m(us)g(forming)e(triangles\).)40 b(Also,)30 b(if)g(a)h(v)-5 b(ariable)29 b Fm(x)2002 3807 y Fh(i)2060 3793 y Fp(o)s(ccurs)h(in)f(clause)i Fm(C)2781 3807 y Fh(j)2848 3793 y Fp(and)e(its)h(negation)p 3520 3743 52 4 v 31 w Fm(x)3572 3807 y Fh(i)0 3906 y Fp(o)s(ccurs)k(in)f(clause)h Fm(C)732 3921 y Fh(j)765 3902 y Ff(0)791 3906 y Fp(,)h(w)m(e)g(join)e (the)i(v)m(ertices)f Fm(v)1709 3920 y Fh(i;j)1824 3906 y Fp(and)g Fm(v)2049 3921 y Fh(i;j)2126 3902 y Ff(0)2185 3906 y Fp(b)m(y)g(an)h(edge.)53 b(V)-8 b(erify)33 b(that)i(there)g(is)e (an)0 4019 y(indep)s(enden)m(t)d(set)i(of)h(size)f Fi(\025)27 b Fm(k)36 b Fp(in)30 b Fm(G)1299 4034 y Fh(\036)1378 4019 y Fp(if)h(and)g(only)h(if)f(there)h(is)f(an)h(assignmen)m(t)g (whic)m(h)f(satis\014es)g Fi(\025)d Fm(k)0 4132 y Fp(of)j(the)f (clauses)g(of)h Fm(\036)p Fp(.)41 b(Hence,)31 b(w)m(e)g(ha)m(v)m(e)136 4302 y Fi(\017)46 b Fm(O)s(P)13 b(T)g Fp(\()p Fm(\036)p Fp(\))26 b(=)f(1)h Fi(\))f Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)1184 4317 y Fh(\036)1230 4302 y Fp(\))26 b Fi(\025)f Fm(m)136 4483 y Fi(\017)46 b Fm(O)s(P)13 b(T)g Fp(\()p Fm(\036)p Fp(\))26 b Fi(\024)f Fm(c)h Fi(\))f Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)1178 4498 y Fh(\036)1224 4483 y Fp(\))26 b Fi(\024)f Fm(cm)0 4653 y Fp(where)33 b Fm(c)e(<)g Fp(1)j(is)e(an)i (absolute)f(constan)m(t.)52 b(By)34 b(the)g Fn(PCP)g Fp(Theorem,)g(w)m(e)h(kno)m(w)e(that)h(there)g(exists)g(a)0 4766 y(p)s(olynomial)28 b(time)i(reduction)f(from)h(SA)-8 b(T)30 b(to)h(Gap-3SA)-8 b(T)q(,)31 b(hence,)f(w)m(e)h(get)0 4937 y Fk(Theorem)i(1)0 5050 y Fe(IS)d(is)f(hard)g(to)j(appro)m(ximate) e(within)1347 5014 y Fg(1)p 1347 5029 36 4 v 1349 5081 a Fh(c)1392 5050 y Fe(.)0 5220 y Fp(In)e(fact,)i(w)m(e)f(can)g(amplify) e(this)g(constan)m(t)j(b)m(y)f(a)g(reduction)e(from)h(an)h(instance)f (of)h(IS)f(with)f(a)i(gap)g(of)3545 5184 y Fh(\013)p 3545 5199 46 4 v 3546 5251 a(\014)0 5394 y Fp(to)i(an)f(instance)g(of)h (IS)f(with)f(gap)1187 5293 y Fd(\020)1252 5358 y Fh(\013)p 1252 5373 V 1253 5425 a(\014)1307 5293 y Fd(\021)1361 5315 y Fg(2)1401 5394 y Fp(.)40 b(By)31 b(rep)s(eating)f(an)m(y)g (constan)m(t)i Fm(k)i Fp(times,)c(w)m(e)h(can)f(sho)m(w)1777 5656 y(1)p eop %%Page: 2 2 2 1 bop 3555 100 a Fp(2)0 399 y Fk(Theorem)33 b(2)0 511 y Fe(IS)d(is)f(hard)g(to)j(appro)m(ximate)e(within)e(a)j(factor)g(of) 1777 438 y Fd(\000)1828 476 y Fg(1)p 1828 491 36 4 v 1830 543 a Fh(c)1873 438 y Fd(\001)1915 459 y Fh(k)1988 511 y Fe(for)f(ev)m(ery)h(constan)m(t)h(in)m(teger)f Fm(k)d Fi(\025)d Fp(1)p Fe(.)0 691 y Fk(Pr)n(oof:)36 b Fp(Giv)m(en)24 b(an)g(instance)f Fm(G)j Fp(=)f(\()p Fm(V)5 b(;)15 b(E)5 b Fp(\),)26 b(construct)e(the)g(graph)f Fm(G)2423 658 y Fj(0)2472 691 y Fp(=)i(\()p Fm(V)2677 658 y Fj(0)2700 691 y Fm(;)15 b(E)2812 658 y Fj(0)2836 691 y Fp(\))24 b(where)f Fm(V)3225 658 y Fj(0)3273 691 y Fp(=)i Fm(V)i Fi(\002)7 b Fm(V)0 804 y Fp(and)29 b Fm(E)248 771 y Fj(0)296 804 y Fp(=)c Fi(f)p Fp(\(\()p Fm(u;)15 b(v)s Fp(\))p Fm(;)g Fp(\()p Fm(u)808 771 y Fj(0)835 804 y Fm(;)g(v)922 771 y Fj(0)946 804 y Fp(\)\))30 b Fi(j)f Fp(\()p Fm(u;)15 b(u)1279 771 y Fj(0)1303 804 y Fp(\))26 b Fi(2)f Fm(V)49 b(or)32 b Fp(\()p Fm(v)s(;)15 b(v)1838 771 y Fj(0)1862 804 y Fp(\))26 b Fi(2)f Fm(E)5 b Fi(g)p Fp(.)41 b(Let)30 b Fm(I)i Fi(\022)25 b Fm(V)49 b Fp(b)s(e)29 b(an)g(indep)s(enden)m(t)e(set)j(of)0 917 y Fm(G)p Fp(.)39 b(The)24 b Fm(I)16 b Fi(\002)8 b Fm(I)31 b Fp(is)24 b(an)g(indep)s(enden)m(t)f(set)i(of)f Fm(G)1542 884 y Fj(0)1590 917 y Fp(b)m(y)h(construction.)38 b(Hence)26 b Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)2845 884 y Fj(0)2868 917 y Fp(\))26 b Fi(\025)f Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)p Fp(\))3375 884 y Fg(2)3415 917 y Fp(.)39 b(On)0 1030 y(the)23 b(other)g(hand,)g(let)g Fm(I)796 997 y Fj(0)842 1030 y Fp(b)s(e)f(an)h(optimal)f(indep)s(enden)m(t)e(set)k(of) f Fm(G)2212 997 y Fj(0)2258 1030 y Fp(with)f(v)m(ertices)h(\()p Fm(u)2866 1044 y Fg(1)2906 1030 y Fm(;)15 b(v)2990 1044 y Fg(1)3030 1030 y Fp(\))p Fm(;)g Fi(\001)g(\001)g(\001)32 b Fm(;)15 b Fp(\()p Fm(u)3369 1045 y Fh(k)3412 1030 y Fm(;)g(v)3496 1045 y Fh(k)3539 1030 y Fp(\).)0 1143 y(By)36 b(construction,)g(the)f(v)m(ertices)h Fm(u)1247 1157 y Fg(1)1287 1143 y Fm(;)15 b Fi(\001)g(\001)g(\001)31 b Fm(;)15 b(u)1555 1158 y Fh(k)1634 1143 y Fp(and)34 b Fm(v)1859 1157 y Fg(1)1899 1143 y Fm(;)15 b Fi(\001)g(\001)g(\001)32 b Fm(;)15 b(v)2160 1158 y Fh(k)2238 1143 y Fp(are)36 b(indep)s(enden)m(t)d(sets)i(in)f Fm(G)p Fp(.)56 b(Hence)0 1256 y(eac)m(h)24 b(con)m(tains)f(at)h(most)f Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)p Fp(\))23 b(distinct)f(v)m(ertices,)j(and)d (therefore)h Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)2766 1223 y Fj(0)2790 1256 y Fp(\))26 b Fi(\024)f Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)p Fp(\))3297 1223 y Fg(2)3337 1256 y Fp(.)38 b(Th)m(us)0 1369 y(w)m(e)31 b(ha)m(v)m(e)g Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)658 1336 y Fj(0)682 1369 y Fp(\))26 b(=)f Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)p Fp(\))1189 1336 y Fg(2)1229 1369 y Fp(.)41 b(Hence)136 1548 y Fi(\017)46 b Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)p Fp(\))26 b Fi(\025)f Fm(\013n)h Fi(\))f Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)1269 1515 y Fj(0)1292 1548 y Fp(\))26 b Fi(\025)f Fm(\013)1507 1515 y Fg(2)1547 1548 y Fm(n)1602 1515 y Fg(2)136 1733 y Fi(\017)46 b Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)p Fp(\))26 b Fi(\024)f Fm(\014)5 b(n)25 b Fi(\))g Fm(O)s(P)13 b(T)g Fp(\()p Fm(G)1266 1700 y Fj(0)1290 1733 y Fp(\))26 b Fi(\024)f Fm(\014)1503 1700 y Fg(2)1542 1733 y Fm(n)1597 1700 y Fg(2)3532 1733 y Fc(2)141 1913 y Fp(Since)k(a)g(maxim)m(um)g (clique)f(of)i Fm(G)g Fp(is)e(a)i(maxim)m(um)e(indep)s(enden)m(t)f(set) j(of)g(the)g(complemen)m(t)p 3347 1840 72 4 v 29 w Fm(G)q Fp(,)g(the)0 2026 y(same)h(ampli\014cation)d(prop)s(ert)m(y)i(\(and)g (hardness)f(result\))h(is)f(true)h(for)g(Maxim)m(um)g(Clique.)141 2229 y(In)c(the)h(next)f(section,)i(w)m(e)f(de\014ne)f(probabilistic)d (v)m(eri\014ers)i(for)h Fn(NP)h Fp(and)f(state)i(the)e(PCP)g(Theorem)0 2342 y(in)j(terms)h(of)h(existence)g(of)f(e\016cien)m(t)h (probabilistic)c(v)m(eri\014ers.)0 2627 y Fl(2)135 b(The)44 b(PCP)h(Theorem)0 2830 y Fn(NP)38 b Fp(can)g(b)s(e)f(de\014ned)g(as)h (the)g(class)f(of)h(languages)g(that)h(are)f(accepted)h(b)m(y)f(p)s (olynomial)d(time)i(non-)0 2943 y(deterministic)f(T)-8 b(uring)36 b(mac)m(hines.)62 b(Equiv)-5 b(alen)m(tly)36 b(it)h(can)i(b)s(e)e(de\014ned)f(in)h(terms)g(of)h(existence)h(of)f(a)0 3055 y(p)s(olytime)29 b(deterministic)f(v)m(eri\014er)i(that)h(can)f(c) m(hec)m(k)i(mem)m(b)s(ership)c(pro)s(ofs)h(:)0 3235 y Fk(Definition)35 b(2)45 b Fo(A)29 b(language)g Fm(L)f Fo(is)h(in)g Fn(NP)g Fo(if)f(and)i(only)g(if)e(ther)-5 b(e)30 b(exists)f(a)g(deterministic)h(p)-5 b(olynomial)0 3348 y(time)32 b(veri\014er)g Fm(V)52 b Fo(such)32 b(that)i(given)d(an) h(input)h Fm(x)p Fo(,)e(and)i(a)g(pr)-5 b(o)g(of)34 b Fm(\031)h Fo(such)d(that)h Fi(j)p Fm(\031)s Fi(j)26 b Fp(=)f Fi(j)p Fm(x)p Fi(j)3019 3315 y Fh(O)r Fg(\(1\))3201 3348 y Fo(it)32 b(satis\014es)136 3528 y Fi(\017)46 b Fo(Completeness:)e Fm(x)25 b Fi(2)g Fm(L)g Fi(\))g(9)32 b Fm(\031)k Fo(such)c(that)i Fm(V)20 b Fp(\()p Fm(x;)15 b(\031)s Fp(\))27 b(=)e(1)136 3712 y Fi(\017)46 b Fo(Soundness:)d Fm(x)35 b(=)-55 b Fi(2)25 b Fm(L)g Fi(\))h(8)32 b Fm(\031)s Fo(,)g Fm(V)20 b Fp(\()p Fm(x;)15 b(\031)s Fp(\))27 b(=)e(0)141 3916 y(No)m(w)32 b(let)g(us)f(de\014ne)f(probabilistic)e(v)m (eri\014ers)j(that)h(are)g(restricted)f(to)h(lo)s(ok)f(at)i(only)d(a)i (few)f(bits)g(of)0 4029 y(the)g(pro)s(of)e(instead)h(of)g(reading)g (the)g(whole)g(pro)s(of.)0 4208 y Fk(Definition)35 b(3)45 b Fo(A)24 b Fp(\()p Fm(r)s Fp(\()p Fm(n)p Fp(\))p Fm(;)15 b(q)s Fp(\()p Fm(n)p Fp(\)\))p Fo(-r)-5 b(estricte)g(d)26 b(veri\014er)f(is)f(one)h(that)g(is)f(r)-5 b(estricte)g(d)26 b(to)f(using)f(at)h(most)g Fm(r)s Fp(\()p Fm(n)p Fp(\))0 4321 y Fo(r)-5 b(andom)35 b(bits,)e(runs)f(in)h(pr)-5 b(ob)g(abilistic)35 b(p)-5 b(olynomial)35 b(time,)e(and)h(queries)e Fm(q)s Fp(\()p Fm(n)p Fp(\))g Fo(bits)h(fr)-5 b(om)34 b(the)f(pr)-5 b(o)g(of.)0 4524 y Fk(Definition)35 b(4)45 b Fo(A)27 b(language)g Fm(L)g Fo(is)f(in)h Fn(PCP)q Fp(\()p Fm(r)s Fp(\()p Fm(n)p Fp(\))p Fm(;)15 b(q)s Fp(\()p Fm(n)p Fp(\)\))28 b Fo(if)e(and)i(only)g(if)e(ther)-5 b(e)28 b(exists)f(a)g Fp(\()p Fm(r)s Fp(\()p Fm(n)p Fp(\))p Fm(;)15 b(q)s Fp(\()p Fm(n)p Fp(\)\))p Fo(-)0 4637 y(r)-5 b(estricte)g(d)34 b(veri\014er)f(such)g(that)h(given)d(an)j(input)f Fm(x;)47 b Fi(j)p Fm(x)p Fi(j)26 b Fp(=)f Fm(n)32 b Fo(and)i(a)f(pr)-5 b(o)g(of)35 b Fm(\031)s Fo(,)d(the)h(veri\014er)g(satis\014es)136 4817 y Fi(\017)46 b Fo(Completeness:)e Fm(x)25 b Fi(2)g Fm(L)g Fi(\))g(9)32 b Fm(\031)k Fo(such)c(that)i Fp(Pr[)p Fm(V)20 b Fp(\()p Fm(x;)15 b(\031)s Fp(\))27 b(=)e(1])h(=)e(1)136 5002 y Fi(\017)46 b Fo(Soundness:)d Fm(x)35 b(=)-55 b Fi(2)25 b Fm(L)g Fi(\))h(8)32 b Fm(\031)s Fo(,)g Fp(Pr[)p Fm(V)20 b Fp(\()p Fm(x;)15 b(\031)s Fp(\))26 b(=)f(1])h Fi(\024)2010 4966 y Fg(1)p 2010 4981 36 4 v 2010 5033 a(2)0 5181 y Fo(wher)-5 b(e)35 b Fm(V)53 b Fo(ac)-5 b(c)g(epts)35 b(or)f(r)-5 b(eje)g(cts)35 b(on)f(the)f(b)-5 b(asis)35 b(of)e(the)h(bits)g(r)-5 b(e)g(ad)35 b(fr)-5 b(om)35 b(the)f(pr)-5 b(o)g(of)36 b(and)e(the)g(pr)-5 b(ob)g(abilities)0 5294 y(ar)g(e)38 b(c)-5 b(ompute)g(d)40 b(over)e(the)f(choic)-5 b(e)38 b(of)g(r)-5 b(andom)40 b(bits.)56 b(Sinc)-5 b(e)38 b(ther)-5 b(e)38 b(ar)-5 b(e)38 b Fp(2)2555 5261 y Fh(r)r Fg(\()p Fh(n)p Fg(\))2729 5294 y Fo(p)-5 b(ossible)39 b(\\runs")e(of)h(the)0 5407 y(veri\014er)33 b(and)g(every)g(run)f(r)-5 b(e)g(ads)35 b Fm(q)s Fp(\()p Fm(n)p Fp(\))e Fo(bits,)f(one)h(c)-5 b(an)33 b(plac)-5 b(e)34 b(an)f(a)g(priori)h(b)-5 b(ound)33 b Fi(j)p Fm(\031)s Fi(j)26 b(\024)f Fm(q)s Fp(\()p Fm(n)p Fp(\))20 b Fi(\001)h Fp(2)3337 5374 y Fh(r)r Fg(\()p Fh(n)p Fg(\))3473 5407 y Fo(.)p eop %%Page: 3 3 3 2 bop 3555 100 a Fp(3)0 399 y Fn(PCP)35 b(for)g(Graph)g (Non-Isomorphism)g(\(GNI\))141 561 y Fp(Tw)m(o)26 b(graphs)f Fm(G)701 575 y Fg(1)766 561 y Fp(=)g(\()p Fm(V)950 575 y Fg(1)990 561 y Fm(;)15 b(E)1097 575 y Fg(1)1137 561 y Fp(\))p Fm(;)g(G)1283 575 y Fg(2)1349 561 y Fp(=)25 b(\()p Fm(V)1533 575 y Fg(2)1573 561 y Fm(;)15 b(E)1680 575 y Fg(2)1720 561 y Fp(\))26 b(on)g Fm(n)f Fp(v)m(ertices)h(are)h (said)d(to)j(b)s(e)e(isomorphic)f(if)h(there)0 674 y(exists)34 b(a)h(p)s(erm)m(utation)f(in)f Fm(S)1026 688 y Fh(n)1073 674 y Fp(,)j Fm(\031)f Fp(:)e Fm(V)1332 688 y Fg(1)1403 674 y Fi(!)g Fm(V)1580 688 y Fg(2)1654 674 y Fp(suc)m(h)h(that)h(\()p Fm(\031)s Fp(\()p Fm(u)p Fp(\))p Fm(;)15 b(\031)s Fp(\()p Fm(v)s Fp(\)\))35 b Fi(2)d Fm(E)2718 688 y Fg(2)2792 674 y Fp(i\013)i(\()p Fm(u;)15 b(v)s Fp(\))34 b Fi(2)e Fm(E)3308 688 y Fg(1)3347 674 y Fp(.)54 b(Tw)m(o)0 787 y(graphs)39 b(are)i(non-isomorphic)c(if)i(there)h(exists)g(no)g(suc)m (h)f(p)s(erm)m(utation.)69 b(Denote)41 b(isomorphism)c(b)m(y)0 900 y Fm(G)71 914 y Fg(1)136 900 y Fi(\031)25 b Fm(G)303 914 y Fg(2)343 900 y Fp(.)0 1088 y Fk(Theorem)33 b(3)0 1200 y Fm(GN)10 b(I)38 b Fi(2)30 b Fn(PCP)p Fp(\()p Fm(O)s Fp(\()p Fm(n)15 b Fp(log)i Fm(n)p Fp(\))p Fm(;)e Fp(1\))0 1413 y Fk(Pr)n(oof:)48 b Fp(The)34 b(input)f Fm(x)f Fp(=)g(\()p Fm(G)1093 1427 y Fg(1)1133 1413 y Fm(;)15 b(G)1244 1427 y Fg(2)1284 1413 y Fp(\),)36 b(is)e(a)h(pair)e(of)i(graphs)e(on)i Fm(n)f Fp(v)m(ertices.)54 b(Eac)m(h)35 b(bit)e(of)i(the)g(pro)s(of)0 1526 y Fm(\031)41 b Fp(corresp)s(onds)d(to)h(a)g(lab)s(eled)d(graph)i (on)h Fm(n)e Fp(v)m(ertices)j Fm(H)7 b Fp(,)40 b(and)e(the)h(bit)e(is)h (supp)s(osed)e(to)j(b)s(e)f(1)h(or)g(2)0 1639 y(resp)s(ectiv)m(ely)31 b(if)g Fm(H)38 b Fp(is)31 b(isomorphic)f(to)i Fm(G)1434 1653 y Fg(1)1506 1639 y Fp(or)f Fm(G)1689 1653 y Fg(2)1729 1639 y Fp(.)45 b(If)31 b Fm(H)39 b Fp(is)30 b(isomorphic)g(to)i (neither,)g(the)g(bit)e(ma)m(y)j(b)s(e)0 1771 y(arbitrary)-8 b(.)40 b(The)31 b(v)m(eri\014er)e(uses)h Fm(O)s Fp(\()p Fm(n)15 b Fp(log)r(\()p Fm(n)p Fp(\)\))31 b(random)f(bits)f(to)i(c)m (ho)s(ose)h Fm(i)2599 1719 y Fh(R)2553 1771 y Fi( )-15 b(\000)25 b(f)p Fp(1)p Fm(;)15 b Fp(2)p Fi(g)33 b Fp(and)d(to)h(c)m(ho) s(ose)h(a)0 1884 y(random)22 b(p)s(erm)m(utation.)38 b(She)22 b(applies)g(the)h(p)s(erm)m(utation)f(to)i(the)g(v)m(ertices)f (of)h Fm(G)2746 1898 y Fh(i)2798 1884 y Fp(to)g(obtain)e(its)h(random)0 1997 y(isomorphic)32 b(cop)m(y)-8 b(,)37 b(sa)m(y)e Fm(H)7 b Fp(.)53 b(She)34 b(queries)f Fm(\031)k Fp(at)f(the)e(p)s(osition)f (corresp)s(onding)f(to)j Fm(H)41 b Fp(and)34 b(accepts)i(if)0 2110 y(and)30 b(only)f(if)h(the)g(bit)g(queried)f(is)g Fm(i)p Fp(.)136 2410 y Fi(\017)46 b Fp(Completeness:)41 b(Supp)s(ose)29 b Fm(G)1263 2424 y Fg(1)1329 2410 y Fi(6\031)d Fm(G)1497 2424 y Fg(2)1537 2410 y Fp(.)42 b(Since)30 b(ev)m(ery)i(graph)e Fm(H)38 b Fp(is)30 b(isomorphic)e(to)k(either)e Fm(G)3448 2424 y Fg(1)3519 2410 y Fp(or)227 2523 y Fm(G)298 2537 y Fg(2)368 2523 y Fp(but)g(not)h(b)s(oth,)f(w)m(e)h(can)f (construct)h(a)g(pro)s(of)e Fm(\031)34 b Fp(suc)m(h)c(that)h Fn(Pr)p Fp([)p Fm(V)21 b Fp(\(\()p Fm(G)2792 2537 y Fg(1)2832 2523 y Fm(;)15 b(G)2943 2537 y Fg(2)2984 2523 y Fp(\))p Fm(;)g(\031)s Fp(\))26 b(=)f(1])h(=)f(1.)136 2711 y Fi(\017)46 b Fp(Soundness:)38 b(Supp)s(ose)27 b Fm(G)1128 2725 y Fg(1)1194 2711 y Fi(\031)e Fm(G)1361 2725 y Fg(2)1400 2711 y Fp(.)41 b(Since)28 b Fm(H)35 b Fp(could)28 b(\\arise")i(from)e Fm(G)2641 2725 y Fg(1)2710 2711 y Fp(or)h Fm(G)2891 2725 y Fg(2)2960 2711 y Fp(with)e(probabilit)m(y)227 2824 y(1)p Fm(=)p Fp(2)47 b(eac)m(h,)j(no)44 b(matter)i(whic)m(h)e(bit)g (the)h(lo)s(cation)g Fm(H)52 b Fp(con)m(tains,)d(the)c(v)m(eri\014er)f (accepts)i(with)227 2936 y(probabilit)m(y)28 b(exactly)j(1)p Fm(=)p Fp(2.)42 b(Th)m(us)30 b(for)g(an)m(y)g(pro)s(of)g Fm(\031)s Fp(,)h Fn(Pr)p Fp([)p Fm(V)21 b Fp(\(\()p Fm(G)2451 2950 y Fg(1)2491 2936 y Fm(;)15 b(G)2602 2950 y Fg(2)2643 2936 y Fp(\))p Fm(;)g(\031)s Fp(\))26 b(=)f(1])h(=)3132 2901 y Fg(1)p 3132 2916 36 4 v 3132 2968 a(2)3532 2936 y Fc(2)141 3124 y Fp(No)m(w)31 b(w)m(e)g(are)g(ready)f(to)h(state)h (the)e(PCP)g(Theorem)g(:)0 3312 y Fk(Theorem)j(4)i(\()p Fn(PCP)f Fk(Theorem\))0 3425 y Fn(NP)26 b Fp(=)e Fn(PCP)q Fp(\()p Fm(O)s Fp(\(log)r(\()p Fm(n)p Fp(\)\))p Fm(;)15 b(O)s Fp(\(1\)\))141 3637 y(The)45 b(inclusion)d Fn(PCP)p Fp(\()p Fm(O)s Fp(\(log)r(\()p Fm(n)p Fp(\))p Fm(;)15 b(O)s Fp(\(1\)\))52 b Fi(\022)e Fn(NP)c Fp(is)e(easy)i(to)g(see.)86 b(Let)45 b Fm(V)66 b Fp(b)s(e)44 b(a)i(v)m(eri\014er)e(for)0 3750 y Fm(L)25 b Fi(2)g Fm(P)13 b(C)7 b(P)13 b Fp(\()p Fm(O)s Fp(\(log)q(\()p Fm(n)p Fp(\))p Fm(;)i(O)s Fp(\(1\)\).)40 b(The)20 b(pro)s(of)g Fm(\031)j Fp(has)e(at)g(most)g(2)2090 3717 y Fh(O)r Fg(\(log)q(\()p Fh(n)p Fg(\)\))2414 3750 y Fp(bits.)36 b(Hence)22 b(w)m(e)f(can)g(construct)g(a)0 3863 y(p)s(olynomial)h(time)i(deterministic)e(v)m(eri\014er)i Fm(V)1576 3830 y Fj(0)1624 3863 y Fp(b)m(y)g(sim)m(ulating)f(all)g(p)s (ossible)f(coin)i(\015ips)f(and)h(computing)0 3976 y(the)31 b(probabilit)m(y)c(that)k Fm(V)51 b Fp(accepts.)42 b(The)30 b(v)m(eri\014er)f Fm(V)1840 3943 y Fj(0)1893 3976 y Fp(accepts)j(if)d (and)h(only)g(if)f(this)g(probabilit)m(y)f(is)h(1.)141 4089 y(No)m(w)35 b(w)m(e)f(will)d(sho)m(w)j(that)g Fn(NP)d Fi(\022)g Fn(PCP)p Fp(\()p Fm(O)s Fp(\(log)17 b Fm(n)p Fp(\))p Fm(;)e(O)s Fp(\(1\)\))36 b(if)c(and)i(only)e(if)h(there)h(is)f (a)h(reduction)0 4202 y(from)f(an)m(y)h Fn(NP)p Fp(-complete)h (language)f(to)g(Gap-MAX-3SA)-8 b(T.)36 b(Th)m(us)d(the)h(PCP)f (Theorem)g(is)g(same)h(as)0 4315 y(an)c(inappro)m(ximabilit)m(y)d (result)i(for)h(MAX-3SA)-8 b(T.)0 4502 y Fk(Theorem)33 b(5)0 4615 y Fn(NP)d Fp(=)g Fn(PCP)p Fp(\()p Fm(O)s Fp(\(log)r(\()p Fm(n)p Fp(\)\))p Fm(;)15 b(O)s Fp(\(1\)\))33 b Fi(,)d(9)i Fe(a)i(p)s(olytime)e(reduction)g(from)h(an)m(y)g Fn(NP)q Fe(-complete)h(language)0 4728 y Fm(L)c Fe(to)h(MAX-3SA)-8 b(T)r(,)30 b(mapping)f(instances)h Fm(x)g Fe(for)g Fm(L)g Fe(to)h(instances)f Fm(\036)h Fe(for)f(MAX-3SA)-8 b(T)31 b(suc)m(h)f(that,)136 4916 y Fi(\017)46 b Fm(x)25 b Fi(2)g Fm(L)g Fi(\))h Fm(O)s(P)13 b(T)g Fp(\()p Fm(\036)p Fp(\))25 b(=)g(1)136 5103 y Fi(\017)46 b Fm(x)25 b Fi(62)g Fm(L)g Fi(\))h Fm(O)s(P)13 b(T)g Fp(\()p Fm(\036)p Fp(\))25 b Fi(\024)g Fm(c;)107 b(c)26 b(<)f Fp(1)p eop %%Page: 4 4 4 3 bop 3555 100 a Fp(4)0 399 y Fk(Pr)n(oof:)55 b Fp(\()p Fi(\()38 b Fp(:\))94 b(Assume)38 b(that)h(for)e Fm(L)h Fi(2)g Fn(NP)g Fp(there)g(is)f(a)i(reduction)e Fm(f)47 b Fp(as)38 b(in)f(the)h(statemen)m(t)i(of)0 511 y(the)i(theorem)f(to)h (MAX-3SA)-8 b(T)43 b(where)e Fm(f)10 b Fp(\()p Fm(x)p Fp(\))43 b(=)g Fm(\036)p Fp(,)i(and)40 b Fm(\036)i Fp(has)f(v)-5 b(ariables)40 b Fm(Y)2797 525 y Fg(1)2836 511 y Fm(;)15 b Fi(\001)g(\001)g(\001)i Fm(Y)3051 525 y Fh(n)3139 511 y Fp(and)40 b(clauses)0 624 y Fm(C)65 638 y Fg(1)104 624 y Fm(;)15 b Fi(\001)g(\001)g(\001)32 b Fm(;)15 b(C)386 638 y Fh(m)483 624 y Fp(where)30 b Fm(m)25 b Fp(=)g Fm(n)1002 591 y Fh(O)r Fg(\(1\))1152 624 y Fp(.)141 737 y(W)-8 b(e)32 b(sho)m(w)e(that)h Fm(L)f Fp(has)g(a)h(PCP)f(with)f Fm(O)s Fp(\(log)q(\()p Fm(n)p Fp(\)\))i(random)f(bits)f(and)h Fm(O)s Fp(\(1\))h(queries.)40 b(The)30 b(v)m(eri\014er)0 850 y Fm(V)53 b Fp(reads)32 b(the)h(input)e Fm(x)h Fp(and)g(pro)s (duces)g(the)h(form)m(ula)f Fm(\036)p Fp(.)47 b(Using)32 b(log)r(\()p Fm(m)p Fp(\))d(=)g Fm(O)s Fp(\(log)r(\()p Fm(n)p Fp(\)\))k(random)f(bits,)0 963 y(she)25 b(c)m(ho)s(oses)g(a)h (uniformly)c(random)i(clause)g Fm(C)1605 977 y Fh(j)1667 963 y Fp(=)h Fm(Y)1836 930 y Fj(\003)1816 989 y Fh(i)1885 963 y Fi(_)9 b Fm(Y)2028 930 y Fj(\003)2008 991 y Fh(k)2076 963 y Fi(_)g Fm(Y)2219 930 y Fj(\003)2199 991 y Fh(`)2284 963 y Fp(of)25 b Fm(\036)p Fp(,)h(where)e(*)i(denotes)f(the)g(v)-5 b(ariable)0 1076 y(or)28 b(its)f(complemen)m(t.)39 b(The)28 b(pro)s(of)e Fm(\031)31 b Fp(corresp)s(onds)26 b(to)i(an)g(assignmen)m (t)f(to)i(the)e(v)-5 b(ariables.)39 b(The)27 b(v)m(eri\014er)0 1189 y(reads)i(the)g(bits)e(corresp)s(onding)g(to)i(the)g(v)-5 b(ariables)28 b(app)s(earing)f(in)h Fm(C)2380 1203 y Fh(j)2445 1189 y Fp(and)g(accepts)i(if)e(and)g(only)g(if)g(the)0 1302 y(v)-5 b(alues)30 b(satisfy)g Fm(C)620 1316 y Fh(j)656 1302 y Fp(.)41 b(Then)29 b(w)m(e)i(ha)m(v)m(e,)136 1471 y Fi(\017)46 b Fp(Completeness:)h(If)33 b Fm(x)e Fi(2)g Fm(L)p Fp(,)k(there)f(exists)f(a)h(satisfying)f(assignmen)m(t)g(for)h Fm(\036)p Fp(.)51 b(Setting)33 b Fm(\031)k Fp(to)e(the)227 1584 y(satisfying)29 b(assignmen)m(t)i(ensures)e(that)i Fn(Pr)p Fp([)p Fm(V)21 b Fp(\()p Fm(x;)15 b(\031)s Fp(\))26 b(=)f(1])h(=)f(1.)136 1757 y Fi(\017)46 b Fp(Soundness:)60 b(If)40 b Fm(x)j Fi(62)f Fm(L)p Fp(,)i(an)m(y)d(assignmen)m(t)g(to)g (the)g(v)-5 b(ariables)40 b Fm(Y)2544 1771 y Fg(1)2583 1757 y Fm(;)15 b Fi(\001)g(\001)g(\001)32 b Fm(;)15 b(Y)2853 1771 y Fh(n)2941 1757 y Fp(satis\014es)40 b(at)i(most)227 1870 y(a)c(fraction)f Fm(c)h Fp(of)g(the)f(clauses.)62 b(Hence,)40 b(for)d(all)g Fm(\031)s Fp(,)i(since)e Fm(V)57 b Fp(c)m(ho)s(oses)39 b(a)e(clause)h(uniformly)c(at)227 1983 y(random)c Fn(Pr)p Fp([)p Fm(V)21 b Fp(\()p Fm(x;)15 b(\031)s Fp(\))26 b(=)f(1])h Fi(\024)f Fm(c)p Fp(.)141 2152 y(The)32 b(soundness)f(probabilit)m(y)f(can)j(b)s(e)e(ampli\014ed) f(b)m(y)j(a)f(constan)m(t)i(n)m(um)m(b)s(er)d(of)i(rep)s(etitions.)45 b(No)m(w)0 2265 y(w)m(e)31 b(pro)m(v)m(e)g(the)g(other)f(direction.)0 2490 y(\()p Fi(\))44 b Fp(:\))111 b(Assume)44 b(that)g Fm(L)j Fi(2)g Fn(NP)d Fp(and)f(has)h(a)g(PCP)f(using)f Fm(O)s Fp(\(log)r(\()p Fm(n)p Fp(\)\))i(random)f(bits)g(and)g Fm(O)s Fp(\(1\))0 2603 y(queries.)37 b(W)-8 b(e)22 b(\014rst)e(reduce)h Fm(L)g Fp(to)g(an)g(in)m(termediate)g(constrain)m(t)g(satisfaction)g (problem)e(whose)i(v)-5 b(ariables)0 2716 y Fm(Y)53 2730 y Fg(1)92 2716 y Fm(;)15 b Fi(\001)g(\001)g(\001)32 b Fm(;)15 b(Y)362 2735 y Fj(j)p Fh(\031)r Fj(j)490 2716 y Fp(are)43 b(the)f(bits)e(in)h(the)h(pro)s(of)f Fm(\031)s Fp(.)76 b(Consider)40 b(a)i(\014xed)f(random)g(string)g Fm(\034)52 b Fp(used)41 b(b)m(y)h(the)0 2829 y(v)m(eri\014er.)g(Let)31 b Fm(Y)577 2796 y Fh(\034)557 2855 y(i)581 2864 y Fb(1)620 2829 y Fm(;)15 b Fi(\001)g(\001)g(\001)32 b Fm(;)15 b(Y)911 2796 y Fh(\034)890 2855 y(i)914 2863 y Fa(q)985 2829 y Fp(b)s(e)30 b(the)h(query)g(bits)f(\014xed)g(b)m(y)h Fm(\034)10 b Fp(.)43 b(Let)31 b Fm(C)2391 2843 y Fh(\034)2461 2829 y Fp(=)26 b Fm(C)7 b Fp(\()p Fm(Y)2737 2796 y Fh(\034)2718 2855 y(i)2742 2864 y Fb(1)2781 2829 y Fm(;)15 b Fi(\001)g(\001)g(\001) 31 b Fm(;)15 b(Y)3071 2796 y Fh(\034)3050 2855 y(i)3074 2863 y Fa(q)3114 2829 y Fp(\))31 b(denote)h(the)0 2964 y(constrain)m(t)26 b(that)g Fm(V)46 b Fp(tests)27 b(for)f(acceptance.) 41 b(The)26 b(n)m(um)m(b)s(er)e(of)i(constrain)m(ts)g(is)f(2)2750 2931 y Fh(O)r Fg(\(log)q(\()p Fh(n)p Fg(\)\))3079 2964 y Fp(=)g Fm(n)3230 2931 y Fh(O)r Fg(\(1\))3379 2964 y Fp(.)39 b(The)0 3077 y(maximization)33 b(problem)g(is)h(to)h(\014nd)e (an)h(assignmen)m(t)h(to)g(the)f(v)-5 b(ariables)34 b Fm(Y)2640 3091 y Fh(i)2667 3077 y Fp('s)h(whic)m(h)e(maximizes)h(the)0 3190 y(fraction)i(of)h(satis\014ed)f(constrain)m(ts.)59 b(This)35 b(is)g(the)i(same)g(as)g(the)g(problem)e(of)i(constructing)f (a)h(pro)s(of)0 3303 y Fm(\031)d Fp(maximizing)29 b(the)j(probabilit)m (y)d(that)i Fm(V)52 b Fp(accepts.)44 b(The)31 b(completeness)g(and)g (soundness)e(conditions)0 3415 y(guaran)m(tee)136 3584 y Fi(\017)46 b Fp(If)40 b Fm(x)j Fi(2)e Fm(L)p Fp(,)i Fi(9)p Fm(\031)h Fp(suc)m(h)c(that)h Fn(Pr)p Fp([)p Fm(V)21 b Fp(\()p Fm(\031)s Fp(\))43 b(=)f(1])h(=)f(1.)71 b(Hence,)44 b(there)d(is)f(an)g(assignmen)m(t)g(to)i(the)227 3697 y(v)-5 b(ariables)29 b Fm(Y)657 3711 y Fg(1)697 3697 y Fm(;)15 b Fi(\001)g(\001)g(\001)31 b Fm(;)15 b(Y)966 3716 y Fj(j)p Fh(\031)r Fj(j)1083 3697 y Fp(suc)m(h)30 b(that)h(all)e(the)i(constrain)m(ts)f Fm(C)2296 3711 y Fh(\034)2370 3697 y Fp(are)g(satis\014ed.)136 3880 y Fi(\017)46 b Fp(If)d Fm(x)j Fi(62)g Fm(L)p Fp(,)h(for)c(all)f Fm(\031)s Fp(,)k Fn(Pr)q Fp([)p Fm(V)20 b Fp(\()p Fm(\031)s Fp(\))48 b(=)e(1])h Fi(\024)1834 3844 y Fg(1)p 1834 3859 36 4 v 1834 3911 a(2)1879 3880 y Fp(.)79 b(Hence,)48 b(an)m(y)43 b(assignmen)m(t)g(to)h(the)f(v)-5 b(ariables)227 3993 y Fm(Y)280 4007 y Fg(1)319 3993 y Fm(;)15 b Fi(\001)g(\001)g(\001) 32 b Fm(;)15 b(Y)589 4011 y Fj(j)p Fh(\031)r Fj(j)706 3993 y Fp(satis\014es)30 b(at)h(most)1385 3957 y Fg(1)p 1385 3972 V 1385 4024 a(2)1460 3993 y Fp(fraction)f(of)h(the)f (constrain)m(ts.)141 4162 y(W)-8 b(e)42 b(will)d(tak)m(e)j(an)f (instance)g(of)g(this)f(constrain)m(t)h(satisfaction)g(problem)e Fm(I)48 b Fp(and)40 b(map)h(it)f(to)i(an)0 4274 y(instance)30 b(of)h(MAX-3SA)-8 b(T)31 b Fm(\036)g Fp(suc)m(h)f(that)136 4426 y Fi(\017)46 b Fm(O)s(P)13 b(T)g Fp(\()p Fm(I)7 b Fp(\))26 b(=)f(1)h Fi(\))f Fm(O)s(P)13 b(T)g Fp(\()p Fm(\036)p Fp(\))26 b(=)e(1)136 4599 y Fi(\017)46 b Fm(O)s(P)13 b(T)g Fp(\()p Fm(I)7 b Fp(\))26 b Fi(\024)685 4563 y Fg(1)p 685 4578 V 685 4630 a(2)755 4599 y Fi(\))g Fm(O)s(P)13 b(T)g Fp(\()p Fm(\036)p Fp(\))25 b Fi(\024)g Fp(1)c Fi(\000)1565 4563 y Fg(1)p 1493 4578 181 4 v 1493 4633 a Fh(q)r Fg(2)1562 4614 y Fa(q)r Fb(+1)141 4761 y Fp(W)-8 b(e)40 b(can)f(write)f(an)m(y)i (constain)m(t)f Fm(C)7 b Fp(\()p Fm(Y)1483 4728 y Fh(\034)1463 4786 y(i)1487 4795 y Fb(1)1526 4761 y Fm(;)15 b Fi(\001)g(\001)g(\001) 32 b Fm(;)15 b(Y)1816 4728 y Fh(\034)1796 4786 y(i)1820 4794 y Fa(q)1859 4761 y Fp(\))39 b(as)h(a)f(CNF)g(with)e(at)j(most)f(2) 2981 4728 y Fh(q)3058 4761 y Fp(clauses)g(and)f Fm(q)0 4874 y Fp(literals)29 b(in)g(eac)m(h)i(clause.)41 b(W)-8 b(e)32 b(can)e(write)g(the)h(CNF)f(as)h(a)g(3SA)-8 b(T)30 b(form)m(ula)g(b)m(y)g(using)f(at)i(most)g Fm(q)i Fp(extra)0 4986 y(v)-5 b(ariables,)29 b(and)h(with)f(at)i(most)g Fm(q)s Fp(2)1209 4953 y Fh(q)1278 4986 y Fp(clauses)f(in)f(total.)141 5099 y(If)35 b(there)h(is)f(a)h(staisfying)e(assignmen)m(t)i(for)f(the) h Fm(Y)1896 5113 y Fh(i)1924 5099 y Fp('s,)i(then)d(this)g(giv)m(es)h (a)g(satisfying)e(assignmen)m(t)0 5212 y(for)d(the)g(3SA)-8 b(T.)43 b(If)31 b(for)f(an)m(y)i(assignmen)m(t,)f(at)h(least)1820 5177 y Fg(1)p 1820 5192 36 4 v 1820 5244 a(2)1896 5212 y Fp(of)g(the)f(constrain)m(ts)g(are)g(unsatis\014ed,)f(then)h(an)m(y)0 5325 y(assignmen)m(t)f(will)e(satisfy)i(at)h(most)g(1)20 b Fi(\000)1497 5289 y Fg(1)p 1424 5304 181 4 v 1424 5359 a Fh(q)r Fg(2)1493 5340 y Fa(q)r Fb(+1)1645 5325 y Fp(fraction)30 b(of)h(clauses)f(of)g(the)h(3SA)-8 b(T)30 b(form)m(ula)g Fm(\036)p Fp(.)188 b Fc(2)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF