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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.|
|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|
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