Integral Polyhedra Related to Even-Cycle and Even-Cut Matroids
Mathematics of Operations Research
Integral Polyhedra Related to Even Cycle and Even Cut Matroids
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
Discrete Applied Mathematics
| Hi-index | 0.00 |
A binary clutter is the family of odd circuits of a binary matroid, that is, the family of circuits that intersect with odd cardinality a fixed given subset of elements. Let A denote the 0,1 matrix whose rows are the characteristic vectors of the odd circuits. A binary clutter is ideal if the polyhedron $\{ x \geq {\bf 0}: \; Ax \geq {\bf 1} \}$ is integral. Examples of ideal binary clutters are st-paths, st-cuts, T-joins or T-cuts in graphs, and odd circuits in weakly bipartite graphs. In 1977, Seymour [J. Combin. Theory Ser. B, 22 (1977), pp. 289--295] conjectured that a binary clutter is ideal if and only if it does not contain ${\cal{L}}_{F_7}$, ${\cal{O}}_{K_5}$, or $b({\cal{O}}_{K_5})$ as a minor. In this paper, we show that a binary clutter is ideal if it does not contain five specified minors, namely the three above minors plus two others. This generalizes Guenin's characterization of weakly bipartite graphs [J. Combin. Theory Ser., 83 (2001), pp. 112--168], as well as the theorem of Edmonds and Johnson [ Math. Programming, 5 (1973), pp. 88--124] on T-joins and T-cuts.