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

Authors Konstantinos Georgiou, Avner Magen, Iannis Tourlakis

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Konstantinos Georgiou
Avner Magen
Iannis Tourlakis

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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)


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.
  • Semidefinite Programming
  • Vertex Cover
  • Integrality Gap
  • Hypermetric Inequalities


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