Pinning on Tight Cuts: Improved Algorithm and Bounds for Unsplittable Multicommodity Flows in Outerplanar Graphs

The multicommodity flow problem in an undirected capacitated graph $G$ is specified by a set of source-sink pairs with nonnegative demands. A flow is feasible if it routes all demands without exceeding the edge capacities, and it is unsplittable if it routes each demand along a single path. Let $α$ be the smallest value such that the existence of a feasible flow implies the existence of an unsplittable flow that exceeds the edge capacities by at most $+\,α\,d_{\max}$, where $d_{\max}$ is the maximum demand value. Schrijver, Seymour, and Winkler showed that $α\in\left[1.01,\,1.5\right]$ if $G$ is a cycle. These bounds were ultimately improved to $α\in\left[1.1,\,1.3\right]$ by Skutella and Däubel. Recently, Alemán Espinosa and Kumar extended this constant upper bound to the broader class of outerplanar graphs, and showed that if $G$ is outerplanar then $α\le 3.6$. We show that $α\in\left[\tfrac{4}{3},2\right]$ if $G$ is outerplanar. We introduce a novel technique that considers the global parameters of the instance, and that may be useful in other (more general) settings where the cut-condition is sufficient, or nearly sufficient, for the existence of a feasible flow.