MaxKFunction - PCPs and Inapproxiability CIS 6930 Sep 15,...

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Unformatted text preview: PCPs and Inapproxiability CIS 6930 Sep 15, 2009 Lecture MAX k-FUNCTION SAT Lecturer: Dr. My T. Thai Scribe: Thang N. Dinh Problem 1 Given n Boolean variables x 1 ,x 2 ,...,x n and m functions f 1 ,...,f m each of which is a function of k of the boolean variables, find a truth assignment to x 1 ,...,x n that maximizes the number of functions satisfied. Here k is a fixed constant (not part of input). Lemma 1 There exists a constant k for which there is a polynomial-time reduc- tion from SAT to MAX k-FUNCTION SAT that transforms a boolean formula to an instance I of MAX k-FUNCTION SAT such that: If is satisfiable, OPT ( I ) = m and If is not satisfiable, then OPT ( I ) < 1 2 m Proof: Note that a E3SAT formula on m clause can be seen as a set of m k-functions where k = 3. Hence, the Theorem 3 on GAP-MAX-E3SAT 1 ,s is a stronger result than this lemma. In other words, the proof presented here is actually a subpart of the proof of the Theorem 3....
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