Tight Gaps for Vertex Cover in the Sherali-Adams SDP Hierarchy

Authors Siavosh Benabbas, Siu On Chan, Konstantinos Georgiou, Avner Magen



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Author Details

Siavosh Benabbas
Siu On Chan
Konstantinos Georgiou
Avner Magen

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Siavosh Benabbas, Siu On Chan, Konstantinos Georgiou, and Avner Magen. Tight Gaps for Vertex Cover in the Sherali-Adams SDP Hierarchy. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 41-54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)
https://doi.org/10.4230/LIPIcs.FSTTCS.2011.41

Abstract

We give the first tight integrality gap for Vertex Cover in the Sherali-Adams SDP system. More precisely, we show that for every \epsilon >0, the standard SDP for Vertex Cover that is strengthened with the level-6 Sherali-Adams system has integrality gap 2-\epsilon. To the best of our knowledge this is the first nontrivial tight integrality gap for the Sherali-Adams SDP hierarchy for a combinatorial problem with hard constraints. For our proof we introduce a new tool to establish Local-Global Discrepancy which uses simple facts from high-dimensional geometry. This allows us to give Sherali-Adams solutions with objective value n(1/2+o(1)) for graphs with small (2+o(1)) vector chromatic number. Since such graphs with no linear size independent sets exist, this immediately gives a tight integrality gap for the Sherali-Adams system for superconstant number of tightenings. In order to obtain a Sherali-Adams solution that also satisfies semidefinite conditions, we reduce semidefiniteness to a condition on the Taylor expansion of a reasonably simple function that we are able to establish up to constant-level SDP tightenings. We conjecture that this condition holds even for superconstant levels which would imply that in fact our solution is valid for superconstant level Sherali-Adams SDPs.
Keywords
  • Vertex Cover
  • Integrality Gap
  • Lift-and-Project systems
  • Linear Programming
  • Semidefinite Programming

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