Approximation resistance from pairwise independent subgroups
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We show that for any eps 0, and positive integers k and q such that q = 2k + 1, given a graph on N vertices that has a q-colorable induced sub graph of (1 - eps)N vertices, it is NP-hard to find an independent set of N/q^{k+1} vertices. This substantially improves upon the work of Dinur et al. [DKPS] who gave a corresponding bound of N/q^2. Our result implies that for any positive integer k, given a graph that has an independent set of approx (2^k + 1)^{-1} fraction of vertices, it is NP-hard to find an independent set of (2k + 1)^{(k+1)} fraction of vertices. This improves on the previous work of Engebretsen and Holmerin [EH] who proved a gap of 2^{-k} vs 2^{-{k choose 2}}, which is best possible using techniques (including those of [EH]) based on the query efficient PCP of Samorodnitsky and Trevisan [3].