On Connectedness of Solutions to Integer Linear Systems

Takasugu Shigenobu; Naoyuki Kamiyama

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arXiv · 2411.19516

On Connectedness of Solutions to Integer Linear Systems

Takasugu Shigenobu·Naoyuki Kamiyama

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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.

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Takasugu Shigenobu, Naoyuki Kamiyama. 2024-11-29. On Connectedness of Solutions to Integer Linear Systems. https://arxiv.org/abs/2411.19516

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