Approximately counting independent sets in dense bipartite graphs via subspace enumeration

TL;DR

This paper presents a randomized algorithm that efficiently approximates the number of independent sets in dense, regular bipartite graphs, extending approximate counting techniques to a low-temperature, dense graph setting.

Contribution

It introduces an FPRAS for #BIS on dense, regular bipartite graphs, applying subspace enumeration to dense graphs in a low-temperature regime, which is a novel approach.

Findings

Provides an FPRAS for #BIS in dense, regular bipartite graphs

Utilizes subspace enumeration exploiting small-set expansion properties

Extends approximate counting methods to dense, low-temperature graph settings

Abstract

We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph -- in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to ``high-temperature'' problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments. The proof exploits the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e. bounded threshold rank), which via subspace enumeration lets us enumerate small cuts efficiently.