Counting Cycles on Planar Graphs in Subexponential Time

TL;DR

This paper introduces a subexponential time algorithm for counting and sampling cycles and self-avoiding walks on triangulated planar graphs, leveraging planar separators and combinatorial analysis, with applications in gerrymandering.

Contribution

It presents the first subexponential algorithm for counting and sampling SAWs on planar graphs, combining separator theorems with Motzkin path analysis.

Findings

Subexponential $2^{O(\sqrt{n})}$ counting algorithm for cycles and SAWs.

Efficient uniform sampling of SAWs in subexponential time.

Application to gerrymandering districting maps.

Abstract

We study the problem of counting all cycles or self-avoiding walks (SAWs) on triangulated planar graphs. We present a subexponential $2^{O(\sqrt{n})}$ time algorithm for this counting problem. Among the technical ingredients used in this algorithm are the planar separator theorem and a delicate analysis using pairs of Motzkin paths and Motzkin numbers. We can then adapt this algorithm to uniformly sample SAWs, in subexponential time. Our work is motivated by the problem of gerrymandered districting maps.