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 U_{2, 4} be the uniform matroid on four elements of rank two, let F₇ be the Fano matroid, let F₇* be the dual of F₇, and let F₇⁺ 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 U_{2, 4}-minor using l, no F₇*-minor using l, and no F₇⁺-minor using l as a series element. Our characterization yields a number of interesting results in combinatorial optimization.