Integer multiflows and metric packings beyond the cut condition

Abstract

Graphs, and more generally matroids, where the simplest possible necessary condition, the ‘Cut Condition’, is also sufficient for multiflow feasibility, have been characterized by Seymour. In this work we exhibit the ‘next’ necessary conditions — there are three of them — and characterize the subclass of matroids where these are also sufficient for multiflow feasibility, or for the existence of integer multiflows in the Eulerian case. Surprisingly, this subclass turns out to properly contain every matroid for which, together with all its minors, the metric packing problem — the ‘polar’ of the multiflow problem — has an integer solution for bipartite data (and a half integer solution in general). We also provide the excluded minor characterization of the corresponding subclass.