A Characterization of Box-Mengerian Matroid Ports

Let M be a matroid on E ∪ {`}, where ` 6∈ E is a distinguished element of M . The `-port of M is the set P = {P : P ⊆ E with P ∪ {`} 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, F7 be the Fano matroid, F ∗ 7 be the dual of F7, and 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 `, no F ∗ 7 -minor using `, and no F 7 -minor using ` as a series element. Our characterization yields a number of interesting results in combinatorial optimization. MSC 2000 subject classification. Primary: 90C10, 90C27, 90C57. OR/MS subject classification. Primary: Programming/graphs.