ETH hardness for densest-k-Subgraph with perfect completeness

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

doi:https://doi.org/10.1137/1.9781611974782.86

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 proceeding › Conference contribution › peer-review

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 language	English
Title of host publication	28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
Editors	Philip N. Klein
Pages	1326-1341
Number of pages	16
ISBN (Electronic)	9781611974782
DOIs	https://doi.org/10.1137/1.9781611974782.86
State	Published - 2017
Externally published	Yes
Event	28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 - Barcelona, Spain Duration: 16 Jan 2017 → 19 Jan 2017

Publication series

Name	Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume	0

Conference

Conference	28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
Country/Territory	Spain
City	Barcelona
Period	16/01/17 → 19/01/17

All Science Journal Classification (ASJC) codes

Software
General Mathematics

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https://doi.org/10.1137/1.9781611974782.86

Cite this

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title = "ETH hardness for densest-k-Subgraph with perfect completeness",

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.",

author = "Mark Braverman and Ko, {Young Kun} and Aviad Rubinstein and Omri Weinstein",

note = "Publisher Copyright: Copyright {\textcopyright} by SIAM.; 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 ; Conference date: 16-01-2017 Through 19-01-2017",

year = "2017",

doi = "https://doi.org/10.1137/1.9781611974782.86",

language = "الإنجليزيّة",

series = "Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms",

pages = "1326--1341",

editor = "Klein, {Philip N.}",

booktitle = "28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017",

}

Braverman, M, Ko, YK, Rubinstein, A & Weinstein, O 2017, ETH hardness for densest-k-Subgraph with perfect completeness. in PN Klein (ed.), 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, vol. 0, pp. 1326-1341, 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, 16/01/17. https://doi.org/10.1137/1.9781611974782.86

ETH hardness for densest-k-Subgraph with perfect completeness. / Braverman, Mark; Ko, Young Kun; Rubinstein, Aviad et al.
28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017. ed. / Philip N. Klein. 2017. p. 1326-1341 (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 0).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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T1 - ETH hardness for densest-k-Subgraph with perfect completeness

AU - Braverman, Mark

AU - Ko, Young Kun

AU - Rubinstein, Aviad

AU - Weinstein, Omri

PY - 2017

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N2 - 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.

AB - 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.

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M3 - منشور من مؤتمر

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BT - 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017

A2 - Klein, Philip N.

T2 - 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017

Y2 - 16 January 2017 through 19 January 2017

ER -

ETH hardness for densest-k-Subgraph with perfect completeness

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