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lec16

# 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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