Combinatorial Algorithms for Edge-Disjoint T-Paths and Integer Free Multiflow

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. The original proof was not constructive, and hence it did not suggest an efficient algorithm. 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 introduce a novel concept of 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 an augmenting walk, we introduce blossoms analogously to the matching algorithm of Edmonds (1965), while it is neither a special case nor a generalization of the present problem. When the search procedure terminates without finding an augmenting walk, the algorithm provides a certificate for the optimality of the current edge-disjoint $T$-paths. Thus the correctness argument of the algorithm serves as an alternative direct proof of Mader's theorem on edge-disjoint $T$-paths. The algorithm runs in $O(|V|\cdot |E|^2)$ time, which is much faster than the best known deterministic algorithm based on a reduction to the linear matroid parity problem. We also present a strongly polynomial algorithm for solving the 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.