Integrality Gaps of Semidefinite Programs for Vertex Cover and Relations to ℓ1 Embeddability of Negative Type Metrics

  • Hamed Hatami
  • Avner Magen
  • Evangelos Markakis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4627)

Abstract

We study various SDP formulations for Vertex Cover by adding different constraints to the standard formulation. We rule out approximations better than \(2-\Omega(\sqrt{1 / \log n})\). We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution embeds into ℓ1 with no distortion, we get an exact relaxation (integrality gap is 1), and on the other hand if the solution is arbitrarily close to being ℓ1 embeddable, the integrality gap is 2 − o(1). Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar to provide a family of simple examples for negative type metrics that cannot be embedded into ℓ1 with distortion better than 8/7 − ε. To this end we prove a new isoperimetric inequality for the hypercube.

Keywords

Triangle Inequality Vertex Cover Isoperimetric Inequality Negative Type Vertex Cover Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Hamed Hatami
    • 1
  • Avner Magen
    • 1
  • Evangelos Markakis
    • 1
  1. 1.Department of Computer Science, University of TorontoCanada

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