Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations

Abstract

We. introduce, a framework for studying semidefiniie programming (SOP) relaxations based on the Lasserre hierarchy in the context of approximation algorithms for combinatorial problems. As an application of our approach, we give, improved approximation algorithms for two problems. We show that for some fixed constant epsiv > 0, given a 3-uniform hypergraph containing an independent set of size (1/2 - epsiv)v, we can find an independent set of size Omega(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">epsiv</sup> ). This improves upon the result of Krivelevich, Nathaniel and Sitdakov, who gave an algorithm finding an independent set of size Omega(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6gamma-3</sup> ) for hypergraphs with an independent set of size gamman (but no guarantee for gamma les 1/2). We also give an algorithm which finds an O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.2072</sup> )-coloring given a 3-colorable graph, improving upon the work of Aurora, Clamtac and Charikar. Our approach stands in contrast to a long series of inapproximability results in the Lovasz Schrijver linear programming (LP) and SDP hierarchies for other problems.