Congestion in planar graphs with demands on faces

TL;DR

This paper presents an algorithm for routing multicommodity flows in planar graphs with congestion proportional to the logarithm of the maximum number of terminals on any face, and proves the limitations of this approach.

Contribution

The paper introduces a new algorithm for multicommodity flow routing in planar graphs with provable congestion bounds and demonstrates the method's optimality.

Findings

Achieves $O(\log k)$ congestion for routing in planar graphs.

Shows the algorithm's congestion bound is close to optimal.

Provides theoretical limits for routing efficiency in planar graphs.

Abstract

We give an algorithm to route a multicommodity flow in a planar graph $G$ with congestion $O(\log k)$, where $k$ is the maximum number of terminals on the boundary of a face, when each demand edge lie on a face of $G$. We also show that our specific method cannot achieve a substantially better congestion.