On Connectedness of Solutions to Integer Linear Systems
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.
Keywords
Cite
@article{arxiv.2411.19516,
title = {On Connectedness of Solutions to Integer Linear Systems},
author = {Takasugu Shigenobu and Naoyuki Kamiyama},
journal= {arXiv preprint arXiv:2411.19516},
year = {2024}
}
Comments
The conference proceedings version of this preprint has appeared in Proceedings of the 16th Annual International Conference on Combinatorial Optimization and Applications (COCOA2023), LNCS 14461, pages 421-433, 2023. This preprint is the submitted version of this paper. Typos and small mistakes were fixed