Sherali-Adams integrality gaps matching the log-density threshold

Eden Chlamtác, Pasin Manurangsi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations

Abstract

The log-density method is a powerful algorithmic framework which in recent years has given rise to the best-known approximations for a variety of problems, including Densest-κ-Subgraph and Small Set Bipartite Vertex Expansion. These approximations have been conjectured to be optimal based on various instantiations of a general conjecture: That it is hard to distinguish a fully random combinatorial structure from one which contains a similar planted sub-structure with the same "log-density". We bolster this conjecture by showing that in a random hypergraph with edge probability n-α, (log n) rounds of Sherali-Adams cannot rule out the existence of a κ-subhypergraph with edge density κ-α-o(1), for any k and α. This holds even when the bound on the objective function is lifted. This gives strong integrality gaps which exactly match the gap in the above distinguishing problems, as well as the best-known approximations, for Densest κ-Subgraph, Smallest p-Edge Subgraph, their hypergraph extensions, and Small Set Bipartite Vertex Expansion (or equivalently, Minimum p-Union). Previously, such integrality gaps were known only for Densest κ-Subgraph for one specific parameter setting.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018
EditorsEric Blais, Jose D. P. Rolim, David Steurer, Klaus Jansen
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print)9783959770859
DOIs
StatePublished - 1 Aug 2018
Event21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 - Princeton, United States
Duration: 20 Aug 201822 Aug 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume116
ISSN (Print)1868-8969

Conference

Conference21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018
Country/TerritoryUnited States
CityPrinceton
Period20/08/1822/08/18

Keywords

  • Approximation algorithms
  • Densest κ-Subgraph
  • Integrality gaps
  • Lift-and-project
  • Log-density

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