A characterization of unimodular hypergraphs with disjoint hyperedges

TL;DR

This paper extends the characterization of total unimodularity from bipartite graphs to disjoint hypergraphs, identifying forbidden structures like odd cycles and odd tree houses, with implications for hypergraph modeling and matrix theory.

Contribution

It provides a new characterization of unimodular hypergraphs with disjoint hyperedges, generalizing the bipartite graph case and resolving a special case of a conjecture on almost totally unimodular matrices.

Findings

Total unimodularity characterized by forbidding odd cycles and odd tree houses.

Extension of unimodularity characterization to disjoint hypergraphs and mixed hypergraphs.

Resolution of a special case of a conjecture on almost totally unimodular matrices.

Abstract

The incidence matrix of a graph is totally unimodular if and only if the graph is bipartite, i.e., it contains no odd cycles. We extend the characterization of total unimodularity to hypergraphs whose hyperedges of size at least four are pairwise disjoint, which we call disjoint hypergraphs. Disjoint hypergraphs have been used to model problems with fairness constraints that ensure balanced representation. We prove that total unimodularity for disjoint hypergraphs is equivalent to forbidding both odd cycles and structures that we call odd tree houses. Our result extends to disjoint mixed hypergraphs, whose incidence matrices have $\{0, \pm1\}$-entries. As a corollary, we resolve a special case of a conjecture on almost totally unimodular matrices, originally posed by Padberg and later modified by Cornu\'ejols and Zuluaga.