Copositivity cuts for improving SDP bounds on the clique number

Authors:
Immanuel M. Bomze;Florian Frommlet;Marco Locatelli
Affiliations:
ISDS, University of Vienna, Vienna, Austria;ISDS, University of Vienna, Vienna, Austria;DII, University of Parma, Parma, Italy
Venue:
Mathematical Programming: Series A and B - Series B - Special Issue: Combinatorial Optimization and Integer Programming
Year:
2010

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Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization

Journal of Global Optimization

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Abstract

Adding cuts based on copositive matrices, we propose to improve Lovász’ bound θ on the clique number and its tightening θ′ introduced by McEliece, Rodemich, Rumsey, and Schrijver. Candidates for cheap and efficient copositivity cuts of this type are obtained from graphs with known clique number. The cost of previously established semidefinite programming bound hierarchies starting with θ′ rapidly increases with the order (and quality requirements). By contrast, the bounds proposed here are relatively cheap in the sense that computational effort is comparable to that required for θ′.