Characterizing unimodular laminar hypergraphs via forbidden subhypergraphs

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 form a laminar family. Such hypergraphs have been used to model problems with fairness constraints that ensure balanced representation, among other applications. Our main result shows that total unimodularity for laminar hypergraphs is equivalent to forbidding odd cycles and structures that we call avocados and tree houses. As a corollary, we resolve a special case of a conjecture on almost totally unimodular matrices, originally posed by Padberg and later modified by Cornuéjols and Zuluaga. We discuss applications of laminar hypergraphs and connect our results to integer programming with bounded subdeterminants.

44 pages, 8 figures