The RPR ² Rounding Technique for Semidefinite Programs

Abstract

Several combinatorial optimization problems can be approximated using algorithms based on semidefinite programming. In many of these algorithms a semidefinite relaxation of the underlying problem is solved yielding an optimal vector configuration v ₁...v _n. This vector configuration is then rounded into a 0,1 solution. We present a procedure called RPR ² (Random Projection followed by Randomized Rounding) for rounding the solution of such semidefinite programs. We show that the random hyperplane rounding technique introduced by Goemans and Williamson, and its variant that involves outward rotation are both special cases of RPR ². We illustrate the use of RPR ² by presenting two applications. For Max-Bisection we improve the approximation ratio. For Max-Cut, we improve the tradeoff curve (presented by Zwick) that relates the approximation ratio to the size of the maximum cut in a graph.