×

Maximum stable set formulations and heuristics based on continuous optimization. (English) Zbl 1023.90071

Summary: The stability number \(\alpha(G)\) for a given graph \(G\) is the size of a maximum stable set in \(G\). The Lovász theta number provides an upper bound on \(\alpha(G)\) and can be computed in polynomial time as the optimal value of the Lovász semidefinite program. In this paper, we show that restricting the matrix variable in the Lovász 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.

MSC:

90C35 Programming involving graphs or networks
90C22 Semidefinite programming
90C59 Approximation methods and heuristics in mathematical programming
90C27 Combinatorial optimization
90C30 Nonlinear programming

Software:

Max-AO; CirCut
PDFBibTeX XMLCite
Full Text: DOI Link