New classes of facets of the cut polytope and tightness of Imm22 Bell inequalities

Authors:
David Avis;Tsuyoshi Ito
Affiliations:
School of Computer Science, McGill University, 3480 University Street, Montreal, Que., Canada H3A 2A7;Department of Computer Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Venue:
Discrete Applied Mathematics
Year:
2007

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Abstract

The Grishukhin inequality is a facet of the cut polytope CUT"7^@? of the complete graph K"7, for which no natural generalization to a family of inequalities has previously been found. On the other hand, the I"m"m"2"2 Bell inequalities of quantum information theory, found by Collins and Gisin, can be seen as valid inequalities of the cut polytope CUT^@?(@?K"m","m) of the complete tripartite graph @?K"m","m=K"1","m","m. They conjectured that they are facet inducing. We prove their conjecture by relating the I"m"m"2"2 inequalities to a new class of facets of CUT"N^@? that are a natural generalization of the Grishukhin inequality. An important component of the proof is the use of a method called triangular elimination, introduced by Avis, Imai, Ito and Sasaki, for producing facets of CUT^@?(@?K"m","m) from facets of CUT"N^@?.