Finding Maximum Edge-Disjoint Paths Between Multiple Terminals

TL;DR

This paper introduces a new combinatorial algorithm for finding maximum edge-disjoint paths connecting multiple terminals in a multigraph, improving efficiency over previous methods.

Contribution

It presents a deterministic, augmenting path-based algorithm using short augmenting walks and blossoms, with a faster $O(|E|^2)$ runtime and applications to multiflow problems.

Findings

Algorithm finds maximum edge-disjoint T-paths efficiently.

Provides a certificate for optimality and an Edmonds-Gallai decomposition.

Includes a strongly polynomial algorithm for the maximum multiflow problem.

Abstract

Let $G=(V,E)$ be a multigraph with a set $T\subseteq V$ of terminals. A path in $G$ is called a $T$-path if its ends are distinct vertices in $T$ and no internal vertices belong to $T$. In 1978, Mader showed a characterization of the maximum number of edge-disjoint $T$-paths. In this paper, we provide a combinatorial, deterministic algorithm for finding the maximum number of edge-disjoint $T$-paths. The algorithm adopts an augmenting path approach. More specifically, we utilize a new concept of short augmenting walks in auxiliary labeled graphs to capture a possible augmentation of the number of edge-disjoint $T$-paths. To design a search procedure for a short augmenting walk, we introduce blossoms analogously to the matching algorithm of Edmonds (1965). When the search procedure terminates without finding a short augmenting walk, the algorithm provides a certificate for the optimality of the current edge-disjoint $T$-paths. From this certificate, one can obtain the Edmonds--Gallai type decomposition introduced by Seb\H{o} and Szeg\H{o} (2004). The algorithm runs in $O(|E|^2)$ time, which is much faster than the best known deterministic algorithm based on a reduction to linear matroid parity. We also present a strongly polynomial algorithm for the maximum integer free multiflow problem, which asks for a nonnegative integer combination of $T$-paths maximizing the sum of the coefficients subject to capacity constraints on the edges.