Abstract
Recently, a characterization of the Lovász theta number based on convex quadratic programming was established. As a consequence of this formulation, we introduce a new upper bound on the stability number of a graph that slightly improves the theta number. Like this number, the new bound can be characterized as the minimum of a function whose values are the optimum values of convex quadratic programs. This paper is oriented mainly to the following question: how can the new bound be used to approximate the maximum stable set for large graphs? With this in mind we present a two-phase heuristic for the stability problem that begins by computing suboptimal solutions using the new bound definition. In the second phase a multi-start tabu search heuristic is implemented. The results of applying this heuristic to some DIMACS benchmark graphs are reported.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 63, Optimal Control, 2009.
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Cavique, L., Luz, C.J. A heuristic for the stability number of a graph based on convex quadratic programming and tabu search. J Math Sci 161, 944–955 (2009). https://doi.org/10.1007/s10958-009-9613-x
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DOI: https://doi.org/10.1007/s10958-009-9613-x