Packing A-paths in Group-Labelled Graphs via Linear Matroid Parity

Mader’s disjoint S-paths problem is a common generalization of matching and Menger’s disjoint paths problems. Lovász (1980) suggested 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 (2003), which leads to faster algorithms. As a generalization of Mader’s problem, Chudnovsky, Geelen, Gerards, Goddyn, Lohman, and Seymour (2006) introduced a framework of packing non-zero A-paths in group-labelled graphs, and proved a min-max theorem. Chudnovsky, Cunningham, and Geelen (2008) provided an efficient combinatorial algorithm for this generalized problem. On the other hand, Pap (2007) introduced a framework of packing non-returning A-paths as a further genaralization. In this paper, we discuss possible extensions of Schrijver’s reduction technique and the algorithm of Chudnovsky, Cunningham, and Geelen to another framework introduced by Pap (2006), under the name of the subgroup model, which apparently generalizes but in fact is equivalent to packing non-returning A-paths. We provide a necessary and sufficient condition for the groups in question to admit a reduction to the linear matroid parity problem. As a consequence, we give faster algorithms for important special cases of packing non-zero A-paths such as odd-length A-paths. In addition, it turns out that packing non-returning A-paths admits a reduction to the linear matroid parity problem, which leads to its efficient solvability, if and only if the size of the input label set is at most four.