Towards an algebraic approach to the reconfiguration CSP

TL;DR

This paper introduces an algebraic framework using partial operations to analyze the complexity of the reconfiguration CSP, extending known results from Boolean domains to broader settings.

Contribution

It presents a novel algebraic approach employing partial operations, enabling the extension of complexity classifications for RCSP beyond Boolean domains.

Findings

Partial operations effectively characterize tractable RCSP instances.

The algebraic framework generalizes complexity results to non-Boolean domains.

The approach offers new insights into the structure of reconfiguration problems.

Abstract

This paper investigates the reconfiguration variant of the Constraint Satisfaction Problem (CSP), referred to as the Reconfiguration CSP (RCSP). Given a CSP instance and two of its solutions, RCSP asks whether one solution can be transformed into the other via a sequence of intermediate solutions, each differing by the assignment of a single variable. RCSP has attracted growing interest in theoretical computer science, and when the variable domain is Boolean, the computational complexity of RCSP exhibits a dichotomy depending on the allowed constraint types. A notable special case is the reconfiguration of graph homomorphisms -- also known as graph recoloring -- which has been studied using topological methods. We propose a novel algebraic approach to RCSP, inspired by techniques used in classical CSP complexity analysis. Unlike traditional methods based on total operations, our framework employs partial operations to capture a reduction involving equality constraints. This perspective facilitates the extension of complexity results from Boolean domains to more general settings, demonstrating the versatility of partial operations in identifying tractable RCSP instances.