Packing Odd Walks and Trails in Multiterminal Networks

Let $G$ be an undirected network with a distinguished set of terminals $T \subseteq V(G)$ and edge capacities $cap: E(G) \rightarrow \mathbb{R}_+$. By an odd $T$-walk we mean a walk in $G$ (with possible vertex and edge self-intersections) connecting two distinct terminals and consisting of an odd number of edges. Inspired by the work of Schrijver and Seymour on odd path packing for two terminals, we consider packings of odd $T$-walks subject to capacities $cap$. First, we present a strongly polynomial time algorithm for constructing a maximum fractional packing of odd $T$-walks. For even integer capacities, our algorithm constructs a packing that is half-integer. Additionally, if $cap(\delta(v))$ is divisible by 4 for any $v \in V(G) - T$, our algorithm constructs an integer packing. Second, we establish and prove the corresponding min-max relation. Third, if $G$ is inner Eulerian (i.e. degrees of all nodes in $V(G) - T$ are even) and $cap(e) = 2$ for all $e \in E$, we show that there exists an integer packing of odd $T$-trails (i.e. odd $T$-walks with no repeated edges) of the same value as in case of odd $T$-walks, and this packing can be found in polynomial time. To achieve the above goals, we establish a connection between packings of odd $T$-walks and $T$-trails and certain multiflow problems in undirected and bidirected graphs.