Integrality gaps of semidefinite programs for Vertex Cover and relations to ℓ1\ell_1 embeddability of Negative Type metrics

We study various SDP formulations for {\sc Vertex Cover} by adding different constraints to the standard formulation. We show that {\sc Vertex Cover} cannot be approximated better than $2-o(1)$ even when we add the so called pentagonal inequality constraints to the standard SDP formulation, en route answering an open question of Karakostas~\cite{Karakostas}. We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution is an $\ell_1$ metric, we get an exact relaxation (integrality gap is 1), and on the other hand if the solution is arbitrarily close to being $\ell_1$ embeddable, the integrality gap may be as big as $2-o(1)$. Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar \cite{Char02} to provide a family of simple examples for negative type metrics that cannot be embedded into $\ell_1$ with distortion better than 8/7-\eps. To this end we prove a new isoperimetric inequality for the hypercube.Comment: A more complete version. Changed order of results. A complete proof of (current) Theorem

The Lovasz number of random graphs

We study the Lovasz number theta along with two further SDP relaxations theta1, theta1/2 of the independence number and the corresponding relaxations of the chromatic number on random graphs G(n,p). We prove that these relaxations are concentrated about their means Moreover, extending a result of Juhasz, we compute the asymptotic value of the relaxations for essentially the entire range of edge probabilities p. As an application, we give an improved algorithm for approximating the independence number in polynomial expected time, thereby extending a result of Krivelevich and Vu. We also improve on the analysis of an algorithm of Krivelevich for deciding whether G(n,p) is k-colorable

Integrality gaps of semidefinite programs for Vertex Cover and relations to ell1_1 embeddability of negative type metrics

We study various SDP formulations for Vertex Cover by adding different constraints to the standard formulation. We rule out approximations better than 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, of . We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution embeds into &#8467;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 &#8467;1 embeddable, the integrality gap is 2&#8201;&#8722;&#8201;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 &#8467;1 with distortion better than 8/7&#8201;&#8722;&#8201;&#949;. To this end we prove a new isoperimetric inequality for the hypercube. </div

Graphs with tiny vector chromatic numbers and huge chromatic numbers

Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246-265] introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly Δ^(1 - 2/k) colors. Here Δ is the maximum degree in the graph and is assumed to be of the order of n^5 for some 0 < δ < 1. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than n/Δ^(1- 2/k) (and hence cannot be colored with significantly fewer than Δ^(1-2/k) colors). For k = O(log n/log log n) we show vector k-colorable graphs that do not have independent sets of size (log n)^c, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylogn. As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653-750] for this problem

Colouring Semirandom Graphs

This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Peer Reviewe