Packing non-zero A-paths via matroid matching

A -labeled graph is a directed graph G in which each edge is associated with an element of a group by a label function :E(G). For a vertex subset AV(G), a path (in the underlying undirected graph) is called an A-path if its start and end vertices belong to A and does not intersect A in between, and an A-path is called non-zero if the ordered product of the labels along the path is not equal to the identity of . Chudnovsky etal. (2006) introduced the problem of packing non-zero A-paths and gave a minmax formula for characterizing the maximum number of vertex-disjoint non-zero A-paths. In this paper, we show that the problem of packing non-zero A-paths can be reduced to the matroid matching problem on a certain combinatorial matroid, and discuss how to derive the minmax formula based on Lovsz idea of reducing Maders S-paths problem to matroid matching.