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DOI: 10.4230/LIPIcs.FSTTCS.2009.2319

On the Tightening of the Standard SDP for Vertex Cover with $ell_1$ Inequalities

Authors: Konstantinos Georgiou, Avner Magen, and Iannis Tourlakis

Published in: LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

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.

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Konstantinos Georgiou, Avner Magen, and Iannis Tourlakis. On the Tightening of the Standard SDP for Vertex Cover with $ell_1$ Inequalities. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 203-214, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{georgiou_et_al:LIPIcs.FSTTCS.2009.2319,
  author =	{Georgiou, Konstantinos and Magen, Avner and Tourlakis, Iannis},
  title =	{{On the Tightening of the Standard SDP for Vertex Cover with \$ell\underline1\$ 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 =	{Kannan, Ravi and Narayan Kumar, K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.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}
}

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On the Tightening of the Standard SDP for Vertex Cover with $ell_1$ Inequalities

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