Generating facets for the cut polytope of a graph by triangular elimination

  • Authors:

  • David Avis;Hiroshi Imai;Tsuyoshi Ito

  • Affiliations:

  • McGill University, School of Computer Science, 3480 University Street, Montreal, QC, Canada;Graduate School of Information Science and Technology, The University of Tokyo, Department of Computer Science and Japan Sci. and Technol. Agency, ERATO-SORST Quantum Computation and Information P ...;Graduate School of Information Science and Technology, The University of Tokyo, Department of Computer Science, 7-3-1 Hongo, Bunkyo-ku, 113-0033, Tokyo, QC, Japan

  • Venue:
  • Mathematical Programming: Series A and B
  • Year:
  • 2007

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Abstract

The cut polytope of a graph arises in many fields. Although much is known about facets of the cut polytope of the complete graph, very little is known for general graphs. The study of Bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or, more generally, the complete k-partite graph. Lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope. In this paper we introduce a lifting operation, named triangular elimination, applicable to the cut polytope of a wide range of graphs. Triangular elimination is a specific combination of zero-lifting and Fourier–Motzkin elimination using the triangle inequality. We prove sufficient conditions for the triangular elimination of facet inducing inequalities to be facet inducing. The proof is based on a variation of the lifting lemma adapted to general graphs. The result can be used to derive facet inducing inequalities of the cut polytope of various graphs from those of the complete graph. We also investigate the symmetry of facet inducing inequalities of the cut polytope of the complete bipartite graph derived by triangular elimination.