Packing $A$-Paths in Group-Labelled Graphs via Linear Matroid Parity

Abstract

Mader's disjoint ${\cal S}$-paths problem is a common generalization of non-bipartite matching and Menger's disjoint paths problems. Lovasz [J. Combin. Theory Ser. B, 28 (1980), pp. 208--236]) proposed a polynomial-time algorithm for this problem through a reduction to matroid matching. A more direct reduction to the linear matroid parity problem was given later by Schrijver [Combinatorial Optimization: Polyhedra and Efficiency, Springer-Verlag, Berlin, 2003], which led to faster algorithms. As a generalization of Mader's problem, Chudnovsky et al. [Combinatorica, 26 (2006), pp. 521--532] introduced a framework of packing non-zero $A$-paths in group-labelled graphs and proved a min-max theorem. Chudnovsky, Cunningham, and Geelen [Combinatorica, 28 (2008), pp. 145--161] provided an efficient combinatorial algorithm for this generalized problem. On the other hand, Pap [Combinatorica, 27 (2007), pp. 247--251] introduced a framework of packing non-returning $A$-paths as a further generalization. In this paper...