Unformatted text preview: ∀ ρ > 0, GAPCLIQUE 1 ,ρ is NPhard) 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, GAPCLIQUE 1 ,nδ is NPhard. As hinted in the lecture note, you may want to choose k to be supperconstant, then the veriﬁer needs a total of O ( k log n )random bits, then the size of G becomes supperpolynomial. 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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 Spring '08
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 Trigraph, Computational complexity theory, 10 pts, NPcomplete

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