On ideal clutters, metrics and multiflows

Abstract

“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 F be a binary clutter. Then the Cut Condition is sufficient in the underlying matroid, for all F ε F as demand-set, to have a multiflow, if and only if it implies the so called K₅, F₇ and R₁₀-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 ε F as demand-set, to have a multiflow, if and only if it implies the Metric Condition for 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. Sebő)”). 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.