Totally equimodular matrices: decomposition and triangulation
For researchers in combinatorial optimization and polyhedral theory, this paper provides foundational building blocks and structural results for totally equimodular matrices, extending known properties of totally unimodular matrices.
This work introduces a decomposition theorem for totally equimodular matrices of full row rank and uses it to prove that simplicial cones generated by such matrices have Hilbert bases and regular unimodular triangulations in most cases, with a conjecture that remaining cases do not exist.
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.