An algorithm for packing non-zero A -paths in group-labelled graphs

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, Chudn...

An Algorithm for Packing Non-zero A-paths in Group-labeled Graphs

Packing Non-Zero A-Paths In Group-Labelled Graphs

Let G=(V,E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A⊆V . An A-path is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P . (If Γ is not abelian, we sum the labels in their order along the path.) We are interested in the maximum number of ...

Packing Non-Returning A-Paths

Chudnovsky et al. gave a min-max formula for the maximum number of node-disjoint non-zero A-paths in group-labeled graphs [1], which is a generalization of Mader’s theorem on node-disjoint A-paths [3]. Here we present a further generalization with a shorter proof. The main feature of Theorem 2.1 is that parity is “hidden” inside ν̂, which is given by an oracle for non-bipartite matching.

Finding a Path in Group-Labeled Graphs with Two Labels Forbidden

The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to the label constraints in a group-labeled graph, which is a directed graph with a group label on each arc. Recently, paths and cycles in group-labeled graphs have been investigated, such as finding non-zero disjoint paths a...