Chvátal cuts and odd cycle inequalities in quadratic 0-1 optimization

• Published in: SIAM journal on discrete mathematics Vol. 5; no. 2; pp. 163 - 177
• Main Authors: BOROS, E, CRAMA, Y, HAMMER, P. L
• Format: Journal Article Web Resource
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.05.1992; Society for Industrial & Applied Mathematics
• Subjects: Applied sciences; Boolean; Business & economic sciences; Chvátal closure; Chvátal cut; Codes; Exact sciences and technology; Grants; Linear programming; Mathematical functions; Mathematical programming; Mathematics; Mathématiques; maximul cut problem; Méthodes quantitatives en économie & gestion; Operational research and scientific management; Operational research. Management science; Optimization; Physical, chemical, mathematical & earth Sciences; Physique, chimie, mathématiques & sciences de la terre; pseudo-Boolean functions; Quantitative methods in economics & management; Sciences économiques & de gestion; unconstrained quadratic 0-1 programming; Variables; weighted 2-SAT; Polyhedron; Lower bound; Maximum; Unconstrained optimization; Graph cut; Linear programming; Graph theory; Quadratic programming; Cycle(graph); Mathematical programming
• ISSN: 0895-4801; 1095-7146; 1095-7146
• DOI: 10.1137/0405014

In this paper a new lower bound for unconstrained quadratic 0 - 1 minimization is investigated. It is shown that this bound can be computed by solving a linear programming problem of polynomial size in the number of variables; and it is shown that the polyhedron ${\text{S}}^{[3]} $, defined by the constraints of this LP formulation is precisely the first Chvatal closure of the polyhedron associated with standard linearization procedures. By rewriting the quadratic minimization problem as a balancing problem in a weighted signed graph, it can be seen that the polyhedron defined by the odd cycle inequalities is equivalent, in a certain sense, with ${\text{S}}^{[3]} $. As a corollary, a compact linear programming formulation is presented for the maximum cut problem for the case of weakly bipartite graphs.