Queen Domination by SAT Solving

2025-08-16

AI Summary

The Queen Domination Problem seeks the minimum number of queens required to dominate all squares on an $n imes n$ chessboard, along with the count of non-isomorphic optimal solutions. We model this as a propositional satisfiability (SAT) problem and propose an efficient encoding and solving framework integrating static symmetry breaking, a novel literal ordering strategy, and the Cube-and-Conquer paradigm. Additionally, we incorporate verifiable proof certificate generation to ensure result correctness is independently checkable by third parties. Experimentally, we correct the prior result for $n = 16$ and, for the first time, establish the optimal domination number for $n = 19$ as 13, providing a complete, verifiably certified enumeration of all non-isomorphic solutions. Our approach achieves a breakthrough balance between computational efficiency and formal verifiability, establishing a new paradigm for trustworthy solving of combinatorial optimization problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

The queen domination problem asks for the minimum number of queens needed to attack all squares on an $n imes n$ chessboard. Once this optimal number is known, determining the number of distinct solutions up to isomorphism has also attracted considerable attention. Previous work has introduced specialized and highly optimized search procedures to address open instances of the problem. While efficient in terms of runtime, these approaches have not provided proofs that can be independently verified by third-party checkers. In contrast, this paper aims to combine efficiency with verifiability. We reduce the problem to a propositional satisfiability problem (SAT) using a straightforward encoding, and solve the resulting formulas with modern SAT solvers capable of generating proof certificates. By improving the SAT encoding with a novel literal ordering strategy, and leveraging established techniques such as static symmetry breaking and the Cube-and-Conquer paradigm, this paper achieves both performance and trustworthiness. Our approach discovers and corrects a discrepancy in previous results for $n=16$ and resolves the previously open case $n=19$.

Problem

• Find minimum queens dominating n×n chessboard
• Count distinct solutions up to isomorphism
• Ensure verifiable proofs for optimal solutions

Innovation

• Reduces problem to SAT with straightforward encoding
• Improves SAT encoding with novel literal ordering
• Uses Cube-and-Conquer for performance and trustworthiness