Counting Solutions for the N-queens and Latin Square Problems by Efficient Monte Carlo Simulations

TL;DR

This paper introduces an efficient Monte Carlo simulation approach to count solutions of the N-queens and Latin square problems by mapping them to thermodynamic systems and using advanced sampling techniques.

Contribution

It develops a novel Monte Carlo method with collective moves to accurately estimate the number of solutions for large combinatorial systems.

Findings

Able to handle systems with 10^4 degrees of freedom.

Successfully estimated solutions exceeding 10^10000.

Observed finite size effects in Latin squares' heat capacity.

Abstract

We apply Monte Carlo simulations to count the numbers of solutions of two well-known combinatorial problems: the N-queens problem and Latin square problem. The original system is first converted to a general thermodynamic system, from which the number of solutions of the original system is obtained by using the method of computing the partition function. Collective moves are used to further accelerate sampling: swap moves are used in the N-queens problem and a cluster algorithm is developed for the Latin squares. The method can handle systems of $10^4$ degrees of freedom with more than $10^10000$ solutions. We also observe a distinct finite size effect of the Latin square system: its heat capacity gradually develops a second maximum as the size increases.