An Approximate Generalization of the Okamura-Seymour Theorem

TL;DR

This paper extends the Okamura-Seymour theorem to a broader class of planar graphs, demonstrating that the cut-condition guarantees routing a constant fraction of demands through an approximate max flow-min cut theorem.

Contribution

It introduces an approximate generalization of the Okamura-Seymour theorem for multicommodity flows in planar graphs with demands on faces.

Findings

Cut-condition guarantees routing of a constant fraction of demands.

Provides an $L_1$-embedding of planar metrics that preserves face distances.

Establishes an approximate max flow-min cut theorem for generalized demand settings.

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 $\Omega(1)$-fraction of all the demands. To prove this, we give a $L_1$-embedding of the planar metric which approximately preserves distance between all pair of points on the same face.