Maximum stable set formulations and heuristics based on continuous optimization

The stability number alpha(G) for a given graph G is the size of a maximum stable set in G. The Lovasz theta number provides an upper bound on alpha(G) and can be computed in polynomial time as the optimal value of the Lovasz semidefinite program. In this paper, we show that restricting the matrix variable in the Lovasz semidefinite program to be rank-one and rank-two, respectively, yields a pair of continuous, nonlinear optimization problems each having the global optimal value alpha(G). We propose heuristics for obtaining large stable sets in G based on these new formulations and present computational results indicating the effectiveness of the heuristics.