1CS151Complexity TheoryLecture 16May 19, 2011May 19, 20112New topic(s)Optimization problems, Approximation Algorithms, and Probabilistically Checkable ProofsMay 19, 20113Approximation Algorithms• “gap-producing” reductionfrom NP-complete problem L1 to L2noyesL1L2 (min. problem)foptkrkMay 19, 20114Gap producing reductions•r-gap-producing reduction:–f computable in poly time–x L1opt(f(x)) k–x L1opt(f(x)) > rk–for max. problemsuse “k” and “< k/r”•Note: target problem is not a language–promise problem(yes no notall strings)– “promise”: instances always from (yes no) May 19, 20115MAX-k-SAT•Missing link: first gap-producing reduction– history’s guideit should have something to do with SAT•Definition:MAX-k-SAT with gap ε–instance: k-CNF φ–YES: some assignment satisfies allclauses–NO: no assignment satisfies more than (1 –ε) fraction of clausesMay 19, 20116Proof systems viewpoint•MAX-k-SAT with gap εNP-hard for any language L NPproof 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] = 1x L Pr[verifier accepts] ≤ (1 – ε)–can repeat O(1/ε) times for error < ½
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