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URN: urn:nbn:de:0030-drops-23195
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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},