Asymptotically Enumerating Independent Sets in Regular $k$-Partite $k$-Uniform Hypergraphs

TL;DR

This paper extends methods for approximating independent sets from bipartite graphs to regular k-partite k-uniform hypergraphs, providing an efficient, local-structure-based enumeration formula, including a closed-form for linear hypergraphs.

Contribution

It introduces a novel application of polymer model techniques to hypergraphs, enabling asymptotic enumeration of independent sets based on local properties.

Findings

Efficient approximation of independent sets in regular hypergraphs.

Derived a simple closed-form for linear hypergraphs.

Generalizes bipartite graph methods to hypergraph context.

Abstract

The number of independent sets in regular bipartite expander graphs can be efficiently approximated by expressing it as the partition function of a suitable polymer model and truncating its cluster expansion. While this approach has been extensively used for graphs, surprisingly little is known about analogous questions in the context of hypergraphs. In this work, we apply this method to asymptotically determine the number of independent sets in regular $k$-partite $k$-uniform hypergraphs which satisfy natural expansion properties. The resulting formula depends only on the local structure of the hypergraph, making it computationally efficient. In particular, we provide a simple closed-form expression for linear hypergraphs.