On semidefinite programming relaxations for graph coloring and vertex cover

We investigate the power of a strengthened SDP relaxation for graph coloring whose value is equal to a variant of the Lovász ϑ-function. We show families of graphs where the value of the relaxation is 2 + ε for any fixed ε > 0, yet the chromatic number is <i>n</i><sup>Δ</sup> for some fixed Δ > 0, which is a function of ε. This demonstrates the bound provided by the SDP is not strong enough to color a 3-colorable graph with <i>n</i><sup>o(1)</sup> colors.Kleinberg and Goemans considered an SDP relaxation for vertex cover whose value is <i>n</i> - ϑ<inf>1/2</inf> (ϑ<inf>1/2</inf> being the variant of the ϑ-function introduced by Schrijver). They asked whether this is within a ratio of 2 - ε of the optimal vertex cover for any ε > 0. Our construction answers this question negatively.

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