Polyhedra with the Integer Carathéodory Property

A polyhedron P has the Integer Caratheodory Property if the following holds. For any positive integer k and any integer vector [email protected]?kP, there exist affinely independent integer vectors x"1,...,x"[email protected]?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.

[1]  Dion Gijswijt,et al.  Integer Decomposition for Polyhedra Defined by Nearly Totally Unimodular Matrices , 2005, SIAM J. Discret. Math..

[2]  Kathie Cameron,et al.  Coflow polyhedra , 1992, Discret. Math..

[3]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[4]  Colin McDiarmid,et al.  Integral decomposition in polyhedra , 1983, Math. Program..

[5]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[6]  藤重 悟.  Submodular functions and optimization , 1991 .

[7]  William H. Cunningham,et al.  Testing membership in matroid polyhedra , 1984, J. Comb. Theory, Ser. B.

[8]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[9]  William J. Cook,et al.  An integer analogue of Carathéodory's theorem , 1986, J. Comb. Theory, Ser. B.

[10]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[11]  L. Trotter,et al.  Integer Rounding and Polyhedral Decomposition for Totally Unimodular Systems , 1978 .

[12]  Alexander Martin,et al.  A counterexample to an integer analogue of Carathéodory's theorem , 1999 .

[13]  András Sebö,et al.  Hilbert Bases, Caratheodory's Theorem and Combinatorial Optimization , 1990, IPCO.

[14]  Brahim Chaourar,et al.  On Greedy Bases Packing in Matroids , 2002, Eur. J. Comb..

[15]  José Coelho de Pina,et al.  Improved bound for the Carathéodory rank of the bases of a matroid , 2003, J. Comb. Theory, Ser. B.

[16]  András Sebő,et al.  Path Partitions, Cycle Covers and Integer Decomposition , 2009 .

[17]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.