# Counting independent sets in unbalanced bipartite graphs

## TL;DR

This paper presents a fully polynomial-time approximation scheme for counting independent sets in unbalanced bipartite graphs with degree or fugacity imbalance, using cluster expansion truncation and establishing exponential decay of correlations.

## Contribution

It introduces a novel FPTAS for the hard-core model on unbalanced bipartite graphs and connects cluster expansion techniques with decay of correlations.

## Key findings

- FPTAS for the hard-core model in unbalanced bipartite graphs
- Exponential decay of correlations established for these graphs
- Applicable to biregular graphs with specific degree conditions

## Abstract

We give an FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides $(L,R)$ of the bipartition. This includes, among others, the biregular case when $\lambda=1$ (approximating the number of independent sets of $G$) and $\Delta_R \geq 7\Delta_L \log(\Delta_L)$. Our approximation algorithm is based on truncating the cluster expansion of a polymer model partition function that expresses the hard-core partition function in terms of deviations from independent sets that are empty on one side of the bipartition. As a consequence of the method, we also prove that the hard-core model on such graphs exhibits exponential decay of correlations by utilizing connections between the cluster expansion and joint cumulants.