🤖 AI Summary
This paper studies the Reconfiguration Constraint Satisfaction Problem (RCSP): given a CSP instance and two solutions, determine whether they can be connected via a sequence of intermediate solutions, each differing from its predecessor in at most one variable assignment. Addressing the limitation of existing algebraic approaches—which require total operations and restrict to Boolean domains—we introduce a novel framework based on **partial algebraic operations**, modeling equality constraints as partial-operation identities and thereby relaxing the classical totality assumption. This framework yields the first universal algebraic characterization of RCSP, unifying solvability criteria across diverse settings. It substantially extends known tractable classes—including graph homomorphism reconfiguration—and provides new decidability criteria and complexity-theoretic analysis for reconstruction problems involving equality constraints.
📝 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.