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# ETH hardness for densest-k-Subgraph with perfect completeness

## Author(s): Braverman, Mark; Ko, YK; Rubinstein, A; Weinstein, O

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 Abstract: We show that, assuming the (deterministic) Exponential Time Hypothesis, distinguishing between a graph with an induced k-clique and a graph in which all k-subgraphs have density at most 1- ϵ, requires n (log n) time. Our result essentially matches the quasi-polynomial algorithms of Feige and Seltser [FS97] and Barman [Bar15] for this problem, and is the first one to rule out an additive PTAS for Densest k-Subgraph. We further strengthen this result by showing that our lower bound continues to hold when, in the soundness case, even subgraphs smaller by a near-polynomial factor (ko = k 2 (log n)) are assumed to be at most (1 - ϵ)-dense. Our reduction is inspired by recent applications of the birthday repetition technique [AIM14, BKW15]. Our analysis relies on information theoretical machinery and is similar in spirit to analyzing a parallel repetition of two- prover games in which the provers may choose to answer some challenges multiple times, while completely ignoring other challenges. Publication Date: 16-Jan-2017 Electronic Publication Date: 2017 Citation: Braverman, M, Ko, YK, Rubinstein, A, Weinstein, O. (2017). ETH hardness for densest-k-Subgraph with perfect completeness. 1326 - 1341 Pages: 1326 - 1341 Type of Material: Journal Article Journal/Proceeding Title: Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms Version: Author's manuscript

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