Computing Circle Packing Representations of Planar Graphs

TL;DR

This paper introduces an algorithm based on convex optimization to compute circle packing representations of planar graphs, with efficient expected runtime and approximation guarantees, advancing the algorithmic understanding of these geometric representations.

Contribution

It provides the first convex optimization-based algorithm for primal-dual circle packings of maximal planar graphs, enabling practical computation of circle packings for all planar graphs.

Findings

Expected runtime is O(n   log(R/b5)) for near-accurate solutions

Algorithm applies to all planar graphs via triangulation

Produces solutions within b5 of true representations

Abstract

The Circle Packing Theorem states that every planar graph can be represented as the tangency graph of a family of internally-disjoint circles. A well-known generalization is the Primal-Dual Circle Packing Theorem for 3-connected planar graphs. The existence of these representations has widespread applications in theoretical computer science and mathematics; however, the algorithmic aspect has received relatively little attention. In this work, we present an algorithm based on convex optimization for computing a primal-dual circle packing representation of maximal planar graphs, i.e. triangulations. This in turn gives an algorithm for computing a circle packing representation of any planar graph. Both take $\widetilde{O}(n \log(R/\varepsilon))$ expected run-time to produce a solution that is $\varepsilon$ close to a true representation, where $R$ is the ratio between the maximum and minimum circle radius in the true representation.