An Approximate Generalization of the Okamura-Seymour Theorem

2022/08/01 by Nikhil Kumar, Kumar, Nikhil
Computer Science · Mathematics · #Advanced Graph Theory Research #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2208.00795

openalex publication_date 2022/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem of multicommodity flows in planar graphs. Okamura and Seymour showed that if all the demands are incident on one face, then the cut-condition is sufficient for routing demands. We consider the following generalization of this setting and prove an approximate max flow-min cut theorem: for every demand edge, there exists a face containing both its end points. We show that the cut-condition is sufficient for routing Ω(1)-fraction of all the demands. To prove this, we give a L₁-embedding of the planar metric which approximately preserves distance between all pair of points on the same face.

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