When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2011.41
URN: urn:nbn:de:0030-drops-33299
URL: http://drops.dagstuhl.de/opus/volltexte/2011/3329/
 Go to the corresponding Portal

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

 pdf-format:

### 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.

### BibTeX - Entry

@InProceedings{benabbas_et_al:LIPIcs:2011:3329,
author =	{Siavosh Benabbas and Siu On Chan and Konstantinos Georgiou and Avner Magen},
title =	{{Tight Gaps for Vertex Cover in the Sherali-Adams SDP Hierarchy}},
booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
pages =	{41--54},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-34-7},
ISSN =	{1868-8969},
year =	{2011},
volume =	{13},
editor =	{Supratik Chakraborty and Amit Kumar},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},