Free multiflows in bidirected and skew-symmetric graphs

Abstract

A graph (digraph) $G=(V,E)$ with a set $T\subseteq V$ 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 and Lov\'asz showed that the maximum number of pairwise edge-disjoint $T$-paths in an inner Eulerian graph $G$ is equal to $\frac12\sum_{s\in T} \lambda(s)$, where $\lambda(s)$ is the minimum number of edges whose removal disconnects $s$ and $T-\{s\}$. A similar relation for inner Eulerian digraphs was established by Lomonosov. Considering undirected and directed networks with ``inner Eulerian'' edge capacities, Ibaraki, Karzanov, and Nagamochi 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 $O(VE\log T\log(2+V^2/E))$ for the corresponding capacitated cases. In particular, this improves the known bound 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.