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-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 give an efficient algorithm for finding ...

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

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.

Packing non-zero A-paths in an undirected model of group labeled graphs

Let Γ be an abelian group, and let γ : E(G) → Γ be be a function assigning values in Γ to every edge of a graph G. For a subgraph H of G, let γ(H) = ∑ e∈E(H) γ(e). In this article, we show that there exists a function f(k) such that the following holds. For a set A of vertices of G, an A-path is a path with both endpoints in A and otherwise disjoint from A. Then either there exist k vertex disj...