The $n$-Queens Problem in Higher Dimensions

TL;DR

This paper explores the higher-dimensional n-queens problem, offering new integer programming formulations, computational speedups, and insights into solution properties within combinatorics and optimization.

Contribution

It introduces strengthened integer programming models and demonstrates significant computational improvements for solving the higher-dimensional n-queens problem.

Findings

Achieved 15-70x speedup over previous benchmarks

Proved optimality for several large instances

Presented heuristic methods matching optimal solutions

Abstract

How many mutually non-attacking queens can be placed on a d-dimensional chessboard of size n? The n-queens problem in higher dimensions is a generalization of the well-known n-queens problem. We provide a comprehensive overview of theoretical results, bounds, solution methods, and the interconnectivity of the problem within topics of discrete optimization and combinatorics. We present an integer programming formulation of the n-queens problem in higher dimensions and several strengthenings through additional valid inequalities. Compared to recent benchmarks, we achieve a speedup in computational time between 15-70x over all instances of the integer programs. Our computational results prove optimality of certificates for several large instances. Breaking additional, previously unsolved instances with the proposed methods is likely possible. On the primal side, we further discuss heuristic approaches to constructing solutions that turn out to be optimal when compared to the IP. We conclude with preliminary results on the number and density of the solutions.