Totally equimodular matrices: decomposition and triangulation

TL;DR

This work develops a structural theory for totally equimodular matrices, a natural generalization of totally unimodular matrices in the box-TDI setting. Central to the paper is a decomposition theorem: any full-row-rank TE set is the disjoint union of mutually-TU te-bricks (tu-sets, te-laces, thin/thick te-interlaces), enabling explicit Hilbert bases for te-cones and constructive regular unimodular Hilbert triangulations in many cases. The authors conect TE with important concepts like Hilbert bases, ICP, and RUHT, and they provide detailed proofs, as well as a conjecture about the remaining cases and a comprehensive suite of consequences for cone decompositions and triangulations. The results yield a unified view linking TE, TU, and various combinatorial matrix classes, with implications for box-TDI polyhedra and integer decomposition properties in toric-like settings. Overall, the paper advances both the theoretical foundations and the algorithmic toolkit for handling TE-cones and their triangulations, establishing new decompositions, HB characterizations, and regular unimodular structures in broad settings.

Abstract

Totally equimodular matrices generalize totally unimodular matrices and arise in the context of box-total dual integral polyhedra. This work further explores the parallels between these two classes and introduces foundational building blocks for constructing totally equimodular matrices. Consequently, we present a decomposition theorem for totally equimodular matrices of full row rank. Building on this decomposition theorem, we prove that simplicial cones whose generators form the rows of a totally equimodular matrix sa\-tisfy strong integrality decomposition properties. More precisely, we provide the Hilbert basis for these cones and construct regular unimodular Hilbert triangulations in most cases. We conjecture that cases not covered here do not exist.

Figures (2)

• Figure 1: A totally equimodular matrix $M$ without full row rank, in which the te-laces are $\{1,2,3,4\}$, $\{3,4,5,6\}$, $\{1,3,5\}$, and $\{2,3,6\}$, and pairwise intersect. Note that there are even intersections of size two.
• Figure 2: The regular unimodular Hilbert triangulation in the case $n=4$. We define $m^{ij}=\frac{1}{2}(a^i+a^j)$, for $i\neq j$. Up to symmetry, the cones in the triangulations are generated by the vertices of the colored tetrahedra in each case.

Theorems & Definitions (115)

• Theorem 1.1: Hoffman and Kruskal Hoffman_Kruskal_1956
• Theorem 1.2: Chervet, Grappe, and Robert Chervet_Grappe_Robert_2021
• Corollary 1.3
• Theorem 3.1: Folklore
• Theorem 3.2: Theorem \ref{['pivotpreserveTE']}
• proof : Sketch
• Lemma 3.3: Lemma \ref{['pivoteqE']}
• Theorem 3.4
• proof
• Theorem 3.5: Theorem \ref{['theorem:decomposition_te_full']}