Free multiflows in bidirected and skew-symmetric graphs

A graph (digraph) G = (V,E) with a set T ⊆ V of terminals is called inner Eulerian if each nonterminal node v has even degree (resp. the numbers of edges entering and leaving v are equal). Cherkassky [1] and Lovász [15] showed that the maximum number of pairwise edge-disjoint T -paths in an inner Eulerian graph G is equal to 1 2 ∑ s∈T λ(s), where λ(s) is the minimum number of edges whose removal disconnects s and T −{s}. A similar relation for inner Eulerian digraphs was established by Lomonosov [14]. Considering undirected and directed networks with “inner Eulerian” edge capacities, Ibaraki, Karzanov, and Nagamochi [10] showed that the problem of finding a maximum integer multiflow (where partial flows connect arbitrary pairs of distinct terminals) is reduced to O(log T ) maximum flow computations and to a number of flow decompositions. In this paper we extend the above max-min relation to inner Eulerian bidirected and skew-symmetric graphs and develop an algorithm of complexity O(V E log T log(2 + V /E)) for the corresponding capacitated cases. In particular, this improves the bound in [10] for digraphs. Our algorithm uses a fast procedure for decomposing a flow with O(1) sources and sinks in a digraph into the sum of one-source-one-sink flows.