紧割集上的固定：外平面图中不可拆分多商品流的改进算法与界
Pinning on Tight Cuts: Improved Algorithm and Bounds for Unsplittable Multicommodity Flows in Outerplanar Graphs
AI总结：
针对外平面图上的不可拆分多商品流问题，通过引入考虑实例全局参数的新技术，将容量超载因子α的上界从3.6改进到2，下界从1.01提升到4/3。
摘要：
无向带容量图$G$中的多商品流问题由一组具有非负需求的源-汇对指定。如果流在不超出边容量的情况下路由所有需求，则它是可行的；如果每条需求沿单一路径路由，则它是不可拆分的。设$α$为最小数值，使得可行流的存在意味着存在一个不可拆分流，其超出边容量的部分最多为$+\,α\,d_{\max}$，其中$d_{\max}$是最大需求值。Schrijver、Seymour和Winkler证明，若$G$是圈，则$α\in\left[1.01,\,1.5\right]$。这些界最终由Skutella和Däubel改进为$α\in\left[1.1,\,1.3\right]$。最近，Alemán Espinosa和Kumar将此常数上界推广到更广泛的外平面图类，并证明若$G$是外平面图，则$α\le 3.6$。我们证明若$G$是外平面图，则$α\in\left[\tfrac{4}{3},2\right]$。我们引入了一种新颖的技术，该技术考虑实例的全局参数，并且可能在其他（更一般的）割条件足以或几乎足以保证可行流存在的情形中有用。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
