License: Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license (CC BY-NC-ND 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2009.2319
URN: urn:nbn:de:0030-drops-23195
URL: https://drops.dagstuhl.de/opus/volltexte/2009/2319/
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### On the Tightening of the Standard SDP for Vertex Cover with $ell_1$ Inequalities

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### Abstract

We show that the integrality gap of the standard SDP for \vc~on instances of $n$ vertices remains $2-o(1)$ even after
the addition of \emph{all} hypermetric inequalities. Our lower bound requires new insights into the structure of SDP
solutions behaving like $\ell_1$ metric spaces when one point is removed. We also show that the addition of all
$\ell_1$ inequalities eliminates any solutions that are not convex combination of integral solutions. Consequently, we
provide the strongest possible separation between hypermetrics and $\ell_1$ inequalities with respect to the
tightening of the standard SDP for \vc.

### BibTeX - Entry

@InProceedings{georgiou_et_al:LIPIcs:2009:2319,
author =	{Konstantinos Georgiou and Avner Magen and Iannis Tourlakis},
title =	{{On the Tightening of the Standard SDP for Vertex Cover with $ell_1$ Inequalities}},
booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
pages =	{203--214},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-13-2},
ISSN =	{1868-8969},
year =	{2009},
volume =	{4},
editor =	{Ravi Kannan and K. Narayan Kumar},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/2319},
URN =		{urn:nbn:de:0030-drops-23195},
doi =		{10.4230/LIPIcs.FSTTCS.2009.2319},
annote =	{Keywords: Semidefinite Programming, Vertex Cover, Integrality Gap, Hypermetric Inequalities}
}


 Keywords: Semidefinite Programming, Vertex Cover, Integrality Gap, Hypermetric Inequalities Collection: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science Issue Date: 2009 Date of publication: 14.12.2009

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