A Customized SAT-based Solver for Graph Coloring

2025-04-07

AI Summary

This paper addresses the Graph Coloring Problem (GCP) by introducing ZykovColor—the first SAT solver specifically designed for GCP using Zykov tree encoding. Methodologically, it integrates three key innovations: (1) a vertex-dominance heuristic for improved variable selection; (2) an incremental bottom-up search framework enabling clause reuse across subproblems; and (3) a pruning mechanism combining efficient clique computation with dynamic lower-bound estimation. The solver is built upon the IPASIR-UP interface, integrating the CaDiCal SAT engine and incorporating a transduction propagator to strengthen constraint propagation. Evaluated on the DIMACS benchmark suite and diverse Erdős–Rényi random graphs—from sparse to dense—ZykovColor consistently outperforms state-of-the-art GCP-dedicated solvers and leading general-purpose SAT approaches, demonstrating substantial gains in both solving efficiency and robustness.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Applications of cryptography

Abstract

We introduce ZykovColor, a novel SAT-based algorithm to solve the graph coloring problem working on top of an encoding that mimics the Zykov tree. Our method is based on an approach of H'ebrard and Katsirelos (2020) that employs a propagator to enforce transitivity constraints, incorporate lower bounds for search tree pruning, and enable inferred propagations. We leverage the recently introduced IPASIR-UP interface for CaDiCal to implement these techniques with a SAT solver. Furthermore, we propose new features that take advantage of the underlying SAT solver. These include modifying the integrated decision strategy with vertex domination hints and using incremental bottom-up search that allows to reuse learned clauses from previous calls. Additionally, we integrate a more efficient clique computation to improve the lower bounds during the search. We validate the effectiveness of each new feature through an experimental analysis. ZykovColor outperforms other state-of-the-art graph coloring implementations on the DIMACS benchmark set. Further experiments on random ErdH{o}s-R'enyi graphs show that our new approach dominates state-of-the-art SAT-based methods for both very sparse and highly dense graphs.

Problem

• Develops SAT-based algorithm for graph coloring
• Enhances search pruning with lower bounds
• Improves efficiency on sparse and dense graphs

Innovation

• SAT-based Zykov tree encoding for graph coloring
• IPASIR-UP interface with CaDiCal SAT solver
• Vertex domination hints and incremental bottom-up search