lec16 - New topic(s) CS151 Complexity Theory Optimization...

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1 CS151 Complexity Theory Lecture 16 May 19, 2011 May 19, 2011 2 New topic(s) Optimization problems, Approximation Algorithms, and Probabilistically Checkable Proofs May 19, 2011 3 Approximation Algorithms • “gap-producing” reduction from NP - complete problem L 1 to L 2 no yes L 1 L 2 (min. problem) f opt k rk May 19, 2011 4 Gap producing reductions • r-gap-producing reduction : – f computable in poly time – x L 1 opt(f(x)) k – x L 1 opt(f(x)) > rk – for max. problems use “ k ” and “ < k/r • Note: target problem is not a language promise problem (yes no not all strings) – “promise”: instances always from (yes no) May 19, 2011 5 MAX-k-SAT • Missing link: first gap-producing reduction – history’s guide it should have something to do with SAT • Definition: MAX-k-SAT with gap ε – instance: k-CNF φ – YES: some assignment satisfies all clauses – NO: no assignment satisfies more than (1 – ε ) fraction of clauses May 19, 2011 6 Proof systems viewpoint • MAX-k-SAT with gap ε NP -hard for any language L NP proof system of form: – given x, compute reduction to MAX-k-SAT: x – expected proof is satisfying assignment for x – verifier picks random clause (“local test”) and checks that it is satisfied by the assignment x L Pr[verifier accepts] = 1 x L Pr[verifier accepts] ≤ (1 – ε) – can repeat O(1/ε) times for error < ½
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2 May 19, 2011 7 Proof systems viewpoint • can think of reduction showing k-SAT NP-hard as designing a proof system for NP in which: – verifier only performs local tests • can think of reduction showing MAX-k-SAT with gap ε NP-hard as designing a proof system for NP in which: – verifier only performs local tests – invalidity of proof* evident all over: “holographic proof” and an fraction of tests notice such invalidity May 19, 2011 8 PCP • Probabilistically Checkable Proof (PCP) permits novel way of verifying proof: – pick random local test – query proof in specified k locations – accept iff passes test • fancy name for a NP-hardness reduction May 19, 2011 9 PCP PCP[r(n),q(n)] : set of languages L with p.p.t. verifier V that has (r, q)-restricted access to a string “proof” – V tosses O(r(n)) coins – V accesses proof in O(q(n)) locations – ( completeness ) x L proof such that Pr[V(x, proof) accepts] = 1 – ( soundness ) x L proof* Pr[V(x, proof*) accepts] ½ May 19, 2011 10 PCP • Two observations: PCP[1, poly n] = NP proof? PCP[log n, 1] NP proof? The PCP Theorem (AS, ALMSS): PCP[log n, 1] = NP . May 19, 2011 11 PCP Corollary : MAX-k-SAT is NP -hard to approximate to within some constant . – using PCP[log n, 1] protocol for, say, VC – enumerate all 2 O(log n) = poly(n) sets of queries – construct a k-CNF φ i for verifier’s test on each • note: k-CNF since function on only k bits – “YES” VC instance all clauses satisfiable – “NO” VC instance every assignment fails to satisfy at least ½ of the φ i fails to satisfy an = (½)2 -k fraction of clauses.
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lec16 - New topic(s) CS151 Complexity Theory Optimization...

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