Umans Complexity Theory Lectures

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1 New topic(s) Umans Complexity Theory Lectures Lecture 15: Approximation Algorithms and Probabilistically Checable Proos (PCPs) Optimization problems,, and Probabilistically Checable Proos 2 Optimization Problems many hard problems (especially NP-hard) are optimization problems e.g. ind shortest TSP tour e.g. ind smallest vertex cover e.g. ind largest clique may be minimization or maximization problem opt = value o optimal solution 3 oten happy with approximately optimal solution warning: lots o heuristics we want approximation algorithm with guaranteed approximation ratio o r meaning: on every input x, output is guaranteed to have value at most r*opt or minimization at least opt/r or maximization 4

2 Example approximation algorithm: Recall: Vertex Cover (VC): given a graph G, what is the smallest subset o vertices that touch every edge? NP-complete Approximation algorithm or VC: pic an edge (x, y), add vertices x and y to VC discard edges incident to x or y; repeat. Claim: approximation ratio is 2. Proo: an optimal VC must include at least one endpoint o each edge considered thereore 2*opt actual 5 6 diverse array o ratios achievable some examples: (min) Vertex Cover: 2 MAX-3-SAT (ind assignment satisying largest # clauses): 8/7 (min) Set Cover: ln n (max) Clique: n/log 2 n (max) Knapsac: (1 + ε) or any ε > 0 (max) Knapsac: (1 + ε) or any ε > 0 called Polymial Time Approximation Scheme (PTAS) algorithm runs in poly time or every ixed ε>0 poor dependence on ε allowed I all NP optimization problems had a PTAS, almost lie P = NP (!) 7 8

3 A job or complexity: How to explain ailure to do better than ratios on previous slide? just lie: how to explain ailure to ind polytime algorithm or SAT... irst guess: probably NP-hard what is needed to show this? gap-producing reduction rom NPcomplete problem L 1 to L 2 gap-producing reduction rom NPcomplete problem L 1 to L 2 L 1 r opt L 2 (min. problem) 9 10 Gap producing reductions r-gap-producing reduction: computable in poly time x L 1 opt((x)) x L 1 opt((x)) > r or max. problems use and < /r Note: target problem is t a language promise problem ( t all strings) promise : instances always rom ( ) L 1 Gap producing reductions L 2 (min.) r Main purpose: r-approximation algorithm or L 2 distinguishes between () and (); can use to decide L 1 NP-hard to approximate to within r L 1 L 2 (max.) /r 11 12

4 Gap preserving reductions gap-producing reduction diicult (more later) but gap-preserving reductions easier Warning: many reductions t gap-preserving r L 1 (min.) r L 2 (min.) 13 Gap preserving reductions Example gap-preserving reduction: reduce MAX--SAT with gap ε to MAX-3-SAT with gap ε constants MAX--SAT is NP-hard to approx. within ε MAX-3-SAT is NP-hard to approx. within ε MAXSNP (PY) a class o problems reducible to each other in this way PTAS or MAXSNP-complete problem i PTAS or all problems in MAXSNP 14 MAX--SAT Missing lin: irst gap-producing reduction history s guide it should have something to do with SAT Deinition: MAX--SAT with gap ε instance: -CNF φ YES: some assignment satisies all clauses NO: assignment satisies more than (1 ε) raction o clauses Proo systems viewpoint -SAT NP-hard or any language L NP proo system o orm: given x, compute reduction to -SAT: ϕ x expected proo is satisying assignment or ϕ x veriier pics random clause ( local test ) and checs that it is satisied by the assignment x L Pr[veriier accepts] = 1 x L Pr[veriier accepts] <

5 Proo systems viewpoint MAX--SAT with gap ε NP-hard or any language L NP proo system o orm: given x, compute reduction to MAX--SAT: ϕ x expected proo is satisying assignment or ϕ x veriier pics random clause ( local test ) and checs that it is satisied by the assignment x L Pr[veriier accepts] = 1 x L Pr[veriier accepts] (1 ε) can repeat O(1/ε) times or error < ½ 17 Proo systems viewpoint can thin o reduction showing -SAT NP-hard as designing a proo system or NP in which: veriier only perorms local tests can thin o reduction showing MAX--SAT with gap ε NP-hard as designing a proo system or NP in which: veriier only perorms local tests invalidity o proo* evident all over: holographic proo and an ε raction o tests tice such invalidity 18 PCP Probabilistically Checable Proo (PCP) permits vel way o veriying proo: pic random local test query proo in speciied locations accept i passes test ancy name or a NP-hardness reduction PCP PCP[r(n),q(n)]: set o languages L with p.p.t. veriier V that has (r, q)-restricted access to a string proo V tosses O(r(n)) coins V accesses proo in O(q(n)) locations (completeness) x L proo such that Pr[V(x, proo) accepts] = 1 (soundness) x L proo* Pr[V(x, proo*) accepts] ½ 19 20

6 PCP Two observations: PCP[1, poly n] = NP proo? PCP[log n, 1] NP proo? The PCP Theorem (AS, ALMSS): PCP[log n, 1] = NP. PCP Corollary: MAX--SAT is NP-hard to approximate to within some constant ε. using PCP[log n, 1] protocol or, say, VC enumerate all 2 O(log n) = poly(n) sets o queries construct a -CNF φ i or veriier s test on each te: -CNF since unction on only bits YES VC instance all clauses satisiable NO VC instance every assignment ails to satisy at least ½ o the φ i ails to satisy an ε = (½)2 - raction o clauses The PCP Theorem Elements o proo: arithmetization o 3-SAT we will do this low-degree test we will state but t prove this sel-correction o low-degree polymials we will state but t prove this proo composition we will describe the idea The PCP Theorem Two major components: NP PCP[log n, polylog n] ( outer veriier ) we will prove this rom scratch, assuming lowdegree test, and sel-correction o low-degree polymials NP PCP[n 3, 1] ( inner veriier ) we will t prove 23 24

7 Proo Composition (idea) NP PCP[log n, polylog n] ( outer veriier ) NP PCP[n 3, 1] ( inner veriier ) composition o veriiers: reormulate outer so that it uses O(log n) random bits to mae 1 query to each o 3 provers replies r 1, r 2, r 3 have length polylog n Key: accept/reject decision computable rom r 1, r 2, r 3 by small circuit C Proo Composition (idea) NP PCP[log n, polylog n] ( outer veriier ) NP PCP[n 3, 1] ( inner veriier ) composition o veriiers (continued): inal proo contains proo that C(r 1, r 2, r 3 ) = 1 or inner veriier s use use inner veriier to veriy that C(r 1,r 2,r 3 ) = 1 O(log n)+polylog n randomness O(1) queries tricy issue: consistency Proo Composition (idea) NP PCP[log n, 1] comes rom repeated composition PCP[log n, polylog n] with PCP[log n, polylog n] yields PCP[log n, polyloglog n] PCP[log n, polyloglog n] with PCP[n 3, 1] yields PCP[log n, 1] many details omitted 27

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