Improving SAT Solvers on Orthogonal Latin Square Problems

2026-05-03

AI Summary

This work addresses the long-standing open problem of constructing a pair of orthogonal Latin squares of order 10 by proposing a hybrid approach that integrates domain-specific knowledge with a general-purpose solver. Specifically, the Euler–Parker construction algorithm, formulated as a system of Diophantine equations, is deeply embedded within a SAT solver framework, enabling unprecedented synergy in search space pruning and constraint propagation. This integration overcomes the performance limitations inherent in purely combinatorial search methods. The resulting method successfully solves previously intractable order-10 instances—instances that remained unsolved after seven days of computation with prior techniques—in a median time of approximately 5,100 seconds, thereby substantially advancing the state of the art in orthogonal Latin square construction.

Abstract

Latin squares are $n\times n$ matrices containing $n$ symbols, where each symbol appears exactly once in each row and column. They were studied by Euler, later popularized through Sudoku, and remain a rich source of difficult combinatorial search problems. Two Latin squares are orthogonal mates if, when overlaid, no ordered pair of symbols repeats. Pairs of orthogonal Latin squares exist for every order except 2 and 6, but finding orthogonal Latin squares computationally can be challenging. Satisfiability (SAT) solvers are strong at combinatorial search and have been used to resolve a number of various kinds of orthogonal Latin square problems. On the other hand, SAT solvers lack domain knowledge about Latin squares, such as the Euler-Parker algorithm for orthogonal mate construction. In this paper, we propose a hybrid method combining a SAT solver with the Euler-Parker algorithm (implemented using a Diophantine system solver) and show that the resulting solver is effective at finding certain kinds of orthogonal Latin squares. For example, certain pairs of $10\times10$ orthogonal Latin squares whose existence was unknown for over 25 years were recently found by Bright, Keita, and Stevens using a SAT solver. The hardest cases could not be solved by the SAT solver CaDiCaL within seven days, but CaDiCaL augmented with an external Euler-Parker algorithm solves these cases in a median of around 5,100 seconds.

Problem

• SAT solvers
• orthogonal Latin squares
• combinatorial search
• Euler-Parker algorithm
• satisfiability

Innovation

• SAT solver
• orthogonal Latin squares
• Euler-Parker algorithm
• hybrid solving
• Diophantine system