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Small Chvátal Rank

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Abstract

We propose a variant of the Chvátal-Gomory procedure that will produce a sufficient set of facet normals for the integer hulls of all polyhedra {x : A x ≤ b} as b varies. The number of steps needed is called the small Chvátal rank (SCR) of A. We characterize matrices for which SCR is zero via the notion of supernormality which generalizes unimodularity. SCR is studied in the context of the stable set problem in a graph, and we show that many of the well-known facet normals of the stable set polytope appear in at most two rounds of our procedure. Our results reveal a uniform hypercyclic structure behind the normals of many complicated facet inequalities in the literature for the stable set polytope. Lower bounds for SCR are derived both in general and for polytopes in the unit cube.

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Correspondence to Tristram Bogart.

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All authors were partially supported by NSF grant DMS-0401047 and the Robert R. and Elaine K. Phelps Endowment at the University of Washington.

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Bogart, T., Raymond, A. & Thomas, R. Small Chvátal Rank. Math. Program. 124, 45–68 (2010). https://doi.org/10.1007/s10107-010-0370-x

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  • DOI: https://doi.org/10.1007/s10107-010-0370-x

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