INTEGRALITY GAPS OF SEMIDEFINITE PROGRAMS FOR VERTEX COVER AND RELATIONS TO e_1 EMBEDDABILITY OF NEGATIVE TYPE METRICS

摘要

We study various semidefinite programming (SDP) formulations for Vertex Cover by adding different constraints to the standard formulation. We show that Vertex Cover cannot be approximated better than 2 - O((log log n/log n)~(1/2)) even when we add the so-called pentagonal inequality constraints to the standard SDP formulation, and thus almost meet the best upper bound known due to Karakostas [Proceedings of the 32nd International Colloquium on Automata, Languages and Programming, 2005], of 2 - Ω((1/ log n)~(1/2)). We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution embeds into e_1 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 e_1 embeddable, the integrality gap is 2 - o(1). Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar [SODA '02: Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2002, pp. 616-620] to provide a family of simple examples for negative type metrics that cannot be embedded into e_1 with distortion better than 8/7 - ∈. To this end we prove a new isoperimetric inequality for the hypercube.