ETH hardness for densest-k-Subgraph with perfect completeness

Mark Braverman, Young Kun Ko, Aviad Rubinstein, Omri Weinstein

Research output: Chapter in Book/Report/Conference proceedingConference contribution

15 Scopus citations

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.

Original languageEnglish (US)
Title of host publication28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
EditorsPhilip N. Klein
PublisherAssociation for Computing Machinery
Pages1326-1341
Number of pages16
ISBN (Electronic)9781611974782
DOIs
StatePublished - Jan 1 2017
Event28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 - Barcelona, Spain
Duration: Jan 16 2017Jan 19 2017

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume0

Other

Other28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
CountrySpain
CityBarcelona
Period1/16/171/19/17

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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  • Cite this

    Braverman, M., Ko, Y. K., Rubinstein, A., & Weinstein, O. (2017). ETH hardness for densest-k-Subgraph with perfect completeness. In P. N. Klein (Ed.), 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 (pp. 1326-1341). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 0). Association for Computing Machinery. https://doi.org/10.1137/1.9781611974782.86