The no-meet matroid

Abstract

Given a digraph and a subset of vertices representing initial positions, we study the existence problem of infinitely long walks, one starting from each initial position, that never meet. We show that the subsets for which this is possible constitute the independent sets of a matroid. We prove that the independence oracle is polynomial-time. We also provide a more efficient algorithm to compute the size of a basis. Then we focus on the orientation problem of the undirected edges of a mixed graph to either maximize or minimize the rank of a subset. Some NP-hardness results and inapproximability results are proved in the general case. Polynomial-time algorithms are described for subsets of size 1. We also report several connections with other matroid classes. For example, we show that the class of no-meet matroids strictly contains transversal matroids while it is strictly contained inside gammoids. Some extensions and related applications are also discussed.