Enumerating Vertices of $0/1$-Polyhedra associated with $0/1$-Totally Unimodular Matrices

TL;DR

This paper presents an efficient algorithm for enumerating vertices of certain 0/1-polyhedra linked to totally unimodular matrices, leveraging hypergraph transversal decomposition and Seymour's matrix decomposition.

Contribution

It introduces an incremental polynomial time algorithm for vertex enumeration of 0/1-polyhedra associated with totally unimodular matrices, using hypergraph and matrix decomposition techniques.

Findings

Algorithm operates in incremental polynomial time.

Uses hypergraph transversal decomposition for unimodular hypergraphs.

Employs Seymour's decomposition of totally unimodular matrices.

Abstract

We give an incremental polynomial time algorithm for enumerating the vertices of any polyhedron $\mathcal{P}(A,\mathbf{1})=\{x\in\RR^n \mid Ax\geq \b1,~x\geq \b0\}$, when $A$ is a totally unimodular matrix. Our algorithm is based on decomposing the hypergraph transversal problem for unimodular hypergraphs using Seymour's decomposition of totally unimodular matrices, and may be of independent interest.