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# hw1 - ∀ ρ> 0 GAP-CLIQUE 1,ρ is NP-hard using the...

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CIS6930: PCPs and Inapproximability - Homework 1 Due at the beginning of the lecture on 10-13-2009 . No late assignment will be accepted. Do the following 6 required problems. Each problem is 10 pts. Problem 1 . Give a gap preserving reduction from MAX-3SAT(29) to MAX-3SAT(5) with appropriate parameters to show the hardness of the latter problem. Problem 2 . Give a gap preserving reduction from MAX-3SAT(d) to MAX-E3SAT(Ed) (each variable appears exactly d times) to show the hardness of the latter problem, that is, GAP-MAX-E3SAT(E d ) 1 is NP-hard for a constant 0 < ρ < 1 Problem 3 . Prove that NP = PCP 1 , 1 /n [log n, log n ] Problem 4 . Show that if there exists an > 0 for which there is a (1+2 /e - )-approximation algorithm for the metric k -Median problem, then NP DTIME ( n O (log log n ) ). Problem 5 . In the class, we mentioned that ρ > 0, GAP-CLIQUE 1 is NP-hard by run- ning PCP verifier k times. Now, prove that theorem (
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Unformatted text preview: ∀ ρ > 0, GAP-CLIQUE 1 ,ρ is NP-hard) using the powering graph (similar to the one we used to show the hardness of approximation of Independent Set). That is, for a graph G = ( V,E ) and an integer k ≥ 2, you need to deﬁne the k th power of G , G k = ( V ,E ) such that ω ( G k ) = ω ( G ) k where ω ( . ) deﬁnes the size of largest cliques in (.). From there, prove the theorem. Problem 6 . Show that there exits a δ > 0, GAP-CLIQUE 1 ,n-δ is NP-hard. As hinted in the lecture note, you may want to choose k to be supper-constant, then the veriﬁer needs a total of O ( k log n )random bits, then the size of G becomes supper-polynomial. Try to show that we can run the veriﬁer k times by using only O (log n ) + O ( k ) (rather than O ( k log n ))...
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