Unsplittable Multicommodity Flows in Outerplanar Graphs

2025-05-19

AI Summary

This paper studies the Undirected Unsplittable Flow Problem (UFP) on outerplanar graphs: routing each demand along a single path while respecting edge capacities and classical cut constraints. As UFP is NP-hard even on outerplanar graphs, we establish the first constant-factor approximation guarantee for this class: by scaling edge capacities to $frac{18}{5} d_{max}$—where $d_{max}$ denotes the maximum demand—we achieve full demand satisfaction. Technically, our approach integrates structural decomposition of outerplanar graphs, refined analysis of cut conditions, path embedding techniques, and a greedy construction strategy. This overcomes the previously known unbounded capacity blow-up barrier for outerplanar graphs and yields the first constant-capacity relaxation result for UFP on a non-tree graph family.

Technology Category

• Planning, Routing, and Scheduling: Planning under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the problem of multicommodity flows in outerplanar graphs. Okamura and Seymour showed that the cut-condition is sufficient for routing demands in outerplanar graphs. We consider the unsplittable version of the problem and prove that if the cut-condition is satisfied, then we can route each demand along a single path by exceeding the capacity of an edge by no more than $frac{18}{5} cdot d_{max}$, where $d_{max}$ is the value of the maximum demand.

Problem

• Study unsplittable multicommodity flows in outerplanar graphs
• Verify cut-condition suffices for routing demands unsplittably
• Bound edge capacity excess by 18/5 times maximum demand

Innovation

• Unsplit multicommodity flows in outerplanar graphs
• Cut-condition ensures single path routing
• Edge capacity exceeded by 18/5*d_max