On Connectedness of Solutions to Integer Linear Systems

TL;DR

This paper investigates the conditions under which the solution graph of an integer linear system is connected, focusing on the role of elimination orderings of the coefficient matrix and establishing when such orderings are necessary or not.

Contribution

It proves that for matrices with at least four rows and three columns, the existence of an elimination ordering is not necessary for solution graph connectedness, but it is necessary for smaller matrices.

Findings

Existence of an EO is not necessary for matrices with ≥4 rows and ≥3 columns.

Existence of an EO is necessary for matrices with ≤3 rows or ≤2 columns.

The paper clarifies the relationship between matrix structure and solution graph connectivity.

Abstract

An integer linear system (ILS) is a linear system with integer constraints. The solution graph of an ILS is defined as an undirected graph defined on the set of feasible solutions to the ILS. A pair of feasible solutions is connected by an edge in the solution graph if the Hamming distance between them is 1. We consider a property of the coefficient matrix of an ILS such that the solution graph is connected for any right-hand side vector. Especially, we focus on the existence of an elimination ordering (EO) of a coefficient matrix, which is known as the sufficient condition for the connectedness of the solution graph for any right-hand side vector. That is, we consider the question whether the existence of an EO of the coefficient matrix is a necessary condition for the connectedness of the solution graph for any right-hand side vector. We first prove that if a coefficient matrix has at least four rows and at least three columns, then the existence of an EO may not be a necessary condition. Next, we prove that if a coefficient matrix has at most three rows or at most two columns, then the existence of an EO is a necessary condition.