Semidefinite programming relaxations for graph coloring and maximal clique problems

Authors:
Igor Dukanovic;Franz Rendl
Affiliations:
Univerza v Mariboru, Ekonomsko-poslovna fakulteta, Razlagova 14, 2000, Maribor, Slovenia;Universität Klagenfurt, Institut für Mathematik, Universitätsstraße 65-67, 9020, Klagenfurt, Austria
Venue:
Mathematical Programming: Series A and B
Year:
2007

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Abstract

The semidefinite programming formulation of the Lovász theta number does not only give one of the best polynomial simultaneous bounds on the chromatic number χ(G) or the clique number ω(G) of a graph, but also leads to heuristics for graph coloring and extracting large cliques. This semidefinite programming formulation can be tightened toward either χ(G) or ω(G) by adding several types of cutting planes. We explore several such strengthenings, and show that some of them can be computed with the same effort as the theta number. We also investigate computational simplifications for graphs with rich automorphism groups.