A characterization of box 1/d1/d-integral binary clutters

Let Q6 denote the port of the dual Fano matroid F*7 and let Q7 denote the clutter consisting of the circuits of the Fano matroid F7 that contain a given element. Let be a binary clutter on E and let d = 2 be an integer. We prove that all the vertices of the polytope {x E+ | x(C) = 1 for C } n {x | a = x = b} are -integral, for any -integral a, b, if and only if does not have Q6 or Q7 as a minor. This includes the class of ports of regular matroids. Applications to graphs are presented, extending a result from Laurent and Pojiak [7]

A characterization of box-mengerian matroid ports

Let M be a matroid on E ∪ {l}, where l ∉ E is a distinguished element of M. The l-port of M is the set P = {P: P ⊆ E with P ∪ {l} a circuit of M }. Let A be the P-E incidence matrix. Let U2,4 be the uniform matroid on four elements of rank two, let F7 be the Fano matroid, let F*7 be the dual of F7, and let F 7 + be the unique series extension of F7. In this paper, we prove that the system Ax ≥ 1, x ≥ 0 is box-totally dual integral (box-TDI) if and only if M has no U2,4-minor using l, no F*7-minor using l, and no F7 +-minor using l as a series element. Our characterization yields a number of interesting results in combinatorial optimization. © 2008 INFORMS.link_to_subscribed_fulltex