Integrality Gaps of Semidefinite Programs for Vertex Cover and Relations to l1Embeddability of Negative Type Metrics

Authors:
Hamed Hatami;Avner Magen;Evangelos Markakis
Affiliations:
Department of Computer Science, University of Toronto,;Department of Computer Science, University of Toronto,;Department of Computer Science, University of Toronto,
Venue:
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Year:
2007

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Abstract

We study various SDP formulations for Vertex Cover by addingdifferent constraints to the standard formulation. We rule outapproximations better than $2-O(\sqrt{\log \log n / \log n})$ evenwhen we add the so-called pentagonal inequality constraints to thestandard SDP formulation, and thus almost meet the best upper boundknown due to Karakostas, of $2-\Omega(\sqrt{1 / \log n})$. Wefurther show the surprising fact that by strengthening the SDP withthe (intractable) requirement that the metric interpretation of thesolution embeds into l1with no distortion, weget an exact relaxation (integrality gap is 1), and on the otherhand if the solution is arbitrarily close to beingl1embeddable, the integrality gap is 2lo(1). Finally, inspired by the above findings,we use ideas from the integrality gap construction of Charikar toprovide a family of simple examples for negative type metrics thatcannot be embedded into l1with distortionbetter than 8/7 ll. To this end we prove a newisoperimetric inequality for the hypercube.