The Width and Integer Optimization on Simplices With Bounded Minors of the Constraint Matrices

TL;DR

This paper demonstrates polynomial-time computability of the width of certain simplices and introduces algorithms for integer linear optimization on these simplices under bounded minors conditions.

Contribution

It provides new polynomial-time algorithms for width computation and integer optimization on simplices with bounded minors in their constraint matrices.

Findings

Width of simplices can be computed in polynomial time under bounded minors.

Algorithms for integer linear optimization on simplices are developed with quasi-polynomial and polynomial complexity.

Bounded minors condition enables efficient optimization on simplices.

Abstract

In this paper, we will show that the width of simplices defined by systems of linear inequalities can be computed in polynomial time if some minors of their constraint matrices are bounded. Additionally, we present some quasi-polynomial-time and polynomial-time algorithms to solve the integer linear optimization problem defined on simplices minus all their integer vertices assuming that some minors of the constraint matrices of the simplices are bounded.