A Blossom Algorithm for Maximum Edge-Disjoint T-Paths

Let G = (V,E) be a multigraph with a set T ⊆ 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 blossom algorithm of Edmonds (1965) for the matching problem, 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 | · |E|) time, which is much faster than the best known deterministic algorithm based on a reduction to the linear matroid parity problem. ∗Supported by JST CREST, No. JPMJCR14D2. †Department of Mathematical Informatics, University of Tokyo, Tokyo 113-8656, Japan. Supported by Grant-inAid for Scientific Research, No. 17H01699 from JSPS. ‡National Institute of Informatics, Tokyo 101-8430, Japan. Supported by Grant-in-Aid for Scientific Research, No. 18K18004 from JSPS.