On Ideal Clutters, Metrics and Multiflows

Abs t rac t . "Binary clutters" contain various objects studied in Combinatorial Optimization, such as paths, Chinese Postman Tours, multiflows and one-sided circuits on surfaces. Minimax theorems about these can be generalized in terms of ideal binary clutters. Seymour has conjectured a characterization of these, and the goal of the present work is to study this conjecture in terms of multiflows in matroids. Seymour's conjecture is equivalent to the following: Let yr be a binary clutter. Then the Cut Condition is su~cient in the underlying matroid, for all F E :F as demand.set, to have a muitiflow, if and only if it implies the so called Ks, Fr and Rio.conditions. These three conditions are applications of the general "Metric Condition" to particular 0 1 bipartite weightings. In this paper we prove the following weakening of this conjecture: The Cut Condition is sufficient for all F E Jr as demand-set, to have a multiflow, ff and only flit implies the Metric Condition [or every bipartite 0 1 weighting. A special case of this result has been stated as a conjecture in Robertson and Seymour's "Graph Minors" volume, (1991, "Open Problem 11 (A. SebS)'). Using Lehman's theorem on minimal non-ideal clutters, we sharpen the properties of minimally non-ideal clutters for the binary and graphic special cases.