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.

翻译：在一张无向带容量图$G$中，多商品流问题由一组具有非负需求的源-汇对指定。如果一条流能在不超出边容量的情况下路由所有需求，则称其为可行的；如果它沿着单一路径路由每条需求，则称其为不可分割的。设$\alpha$为最小标量，使得存在可行流蕴含存在至多超出边容量$+\,α\,d_{\max}$的不可分割流，其中$d_{\max}$为最大需求值。Schrijver、Seymour和Winkler证明了若$G$为环图，则$\alpha\in\left[1.01,\,1.5\right]$。这些界最终被Skutella和Däubel改进为$\alpha\in\left[1.1,\,1.3\right]$。最近，Alemán Espinosa和Kumar将此常值上界推广至更广泛的外平面图类，并证明了若$G$为外平面图，则$\alpha\le 3.6$。我们证明若$G$为外平面图，则$\alpha\in\left[\tfrac{4}{3},2\right]$。我们引入了一种考虑实例全局参数的新技术，该技术或可应用于其他（更一般）的、割条件对可行流存在性充分（或几乎充分）的设定中。