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.
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Hatami, H., Magen, A., Markakis, E. (2007). Integrality Gaps of Semidefinite Programs for Vertex Cover and Relations to ℓ1 Embeddability of Negative Type Metrics. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_12
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