Multicommodity Flows in Planar Graphs with Demands on Faces

TL;DR

This paper investigates multicommodity flows in planar graphs with demands on faces, establishing a flow-cut gap bound of 3 under specific source-sink arrangements and introducing a novel demand approximation method.

Contribution

It generalizes previous results by showing a bounded flow-cut gap for a new class of demand configurations on planar graphs.

Findings

Flow-cut gap is at most 3 for specified demand arrangements.

Introduces a convex combination approach to approximate face demands.

Extends known results to more general face-based demand scenarios.

Abstract

We consider the problem of multicommodity flows in planar graphs. Seymour showed that if the union of supply and demand graphs is planar, then the cut condition is sufficient for routing demands. Okamura-Seymour showed that if all demands are incident on one face, then again cut condition is sufficient for routing demands. We consider a common generalization of these settings where the end points of each demand are on the same face of the planar graph. We show that if the source sink pairs on each face of the graph are such that sources and sinks appear contiguously on the cycle bounding the face, then the flow cut gap is at most 3. We come up with a notion of approximating demands on a face by convex combination of laminar demands to prove this result.