Burer, Samuel; Monteiro, Renato D. C.; Zhang, Yin Maximum stable set formulations and heuristics based on continuous optimization. (English) Zbl 1023.90071 Math. Program. 94, No. 1 (A), 137-166 (2002). 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. Cited in 24 Documents MSC: 90C35 Programming involving graphs or networks 90C22 Semidefinite programming 90C59 Approximation methods and heuristics in mathematical programming 90C27 Combinatorial optimization 90C30 Nonlinear programming Keywords:Lovász semidefinite program; heuristics; maximum clique; minimum vertex cover; semidefinite relaxation Software:Max-AO; CirCut PDFBibTeX XMLCite \textit{S. Burer} et al., Math. Program. 94, No. 1 (A), 137--166 (2002; Zbl 1023.90071) Full Text: DOI Link