Polyhedra with the Integer Carathéodory Property

  • Authors:
  • Dion Gijswijt;Guus Regts

  • Affiliations:
  • CWI, Amsterdam, The Netherlands and Dep. of Mathematics, Leiden University, The Netherlands;CWI, Amsterdam, The Netherlands

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2012

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Abstract

A polyhedron P has the Integer Caratheodory Property if the following holds. For any positive integer k and any integer vector w@?kP, there exist affinely independent integer vectors x"1,...,x"t@?P and positive integers n"1,...,n"t such that n"1+...+n"t=k and w=n"1x"1+...+n"tx"t. In this paper we prove that if P is a (poly)matroid base polytope or if P is defined by a totally unimodular matrix, then P and projections of P have the Integer Caratheodory Property. For the matroid base polytope this answers a question by Cunningham from 1984.