Trichotomy for the reconfiguration problem of integer linear systems

TL;DR

This paper analyzes the complexity of reconfiguring feasible solutions in integer linear systems, revealing a trichotomy based on a complexity index, with implications for Horn and two-variable-par-inequality systems.

Contribution

It introduces a complexity index-based classification for the reconfiguration problem, establishing new complexity results including coNP and PSPACE classifications.

Findings

Reconfiguration is in constant time if index < 1.

Reconfiguration is coNP-complete when index = 1.

Reconfiguration is PSPACE-complete when index > 1.

Abstract

In this paper, we consider the reconfiguration problem of integer linear systems. In this problem, we are given an integer linear system $I$ and two feasible solutions $\boldsymbol{s}$ and $\boldsymbol{t}$ of $I$, and then asked to transform $\boldsymbol{s}$ to $\boldsymbol{t}$ by changing a value of only one variable at a time, while maintaining a feasible solution of $I$ throughout. $Z(I)$ for $I$ is the complexity index introduced by Kimura and Makino (Discrete Applied Mathematics 200:67--78, 2016), which is defined by the sign pattern of the input matrix. We analyze the complexity of the reconfiguration problem of integer linear systems based on the complexity index $Z(I)$ of given $I$. We then show that the problem is (i) solvable in constant time if $Z(I)$ is less than one, (ii) weakly coNP-complete and pseudo-polynomially solvable if $Z(I)$ is exactly one, and (iii) PSPACE-complete if $Z(I)$ is greater than one. Since the complexity indices of Horn and two-variable-par-inequality integer linear systems are at most one, our results imply that the reconfiguration of these systems are in coNP and pseudo-polynomially solvable. Moreover, this is the first result that reveals coNP-completeness for a reconfiguration problem, to the best of our knowledge.