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Towards an algebraic approach to the reconfiguration CSP

Kei Kimura

TL;DR

The paper introduces an algebraic framework for the reconfiguration CSP (RCSP) based on partial polymorphisms, enabling reductions and tractability results under equality constraints. It defines and analyzes ordered partial Maltsev operations to characterize safely OR-free (SOF) and related tractable RCSP classes, extending known Boolean results to broader domains. Key contributions include a polynomial-time algorithm for RCSP when an ordered partial Maltsev operation is a partial polymorphism, and a detailed study of the invariance properties of various relation classes (SOF, SNF, SCB) and their digraph analogues. The work highlights the potential of combining algebraic and topological perspectives in RCSP complexity, while acknowledging gaps in the theory of partial operations and pp-interpretations.

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.

Towards an algebraic approach to the reconfiguration CSP

TL;DR

The paper introduces an algebraic framework for the reconfiguration CSP (RCSP) based on partial polymorphisms, enabling reductions and tractability results under equality constraints. It defines and analyzes ordered partial Maltsev operations to characterize safely OR-free (SOF) and related tractable RCSP classes, extending known Boolean results to broader domains. Key contributions include a polynomial-time algorithm for RCSP when an ordered partial Maltsev operation is a partial polymorphism, and a detailed study of the invariance properties of various relation classes (SOF, SNF, SCB) and their digraph analogues. The work highlights the potential of combining algebraic and topological perspectives in RCSP complexity, while acknowledging gaps in the theory of partial operations and pp-interpretations.

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.

Paper Structure

This paper contains 14 sections, 35 theorems, 13 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Polymorphism and logical expression (reduction)
  5. Partial polymorphism and logical expression (reduction)
  6. Known results for the Boolean RCSP
  7. Safely OR-free (NAND-free) relations and their extension
  8. Ordered partial Maltsev operation and characterization of $\varGamma_{{\rm SOF}}$
  9. Algorithm for RCSP invariant under an ordered partial Maltsev operation
  10. Relations invariant under some ordered partial Maltsev operation
  11. Relations invariant under some ordered partial Maltsev operation
  12. Digraphs invariant under some ordered partial Maltsev operation
  13. Safely componentwise bijunctive relations
  14. Conclusion

Key Result

Proposition 2.14

$\langle \varGamma \rangle_{\exists,\wedge,=,{\rm \bf f}} = {\rm Inv}({\rm Pol}(\varGamma))$.

Figures (2)

  • Figure 1: The two non-isomorphic orientations of a 4-cycle of algebraic girth zero
  • Figure 2: The orientation of a 3-cycle of algebraic girth one

Theorems & Definitions (82)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6: CSP
  • Example 2.7
  • Definition 2.8: Solution graph
  • Definition 2.9: RCSP
  • Definition 2.10: Operation and polymorphism
  • ...and 72 more