Solving the n-Queens Problem in Higher Dimensions

TL;DR

This paper extends the n-queens problem to higher dimensions, proposing an integer programming approach with enhancements, resulting in significant computational speedups and solving large instances optimally.

Contribution

It introduces a new integer programming formulation for higher-dimensional n-queens and improves computational efficiency over existing methods.

Findings

Achieved 15-70x speedup over previous benchmarks.

Proved optimality for several large instances.

Potential to solve more previously unsolved instances.

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 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.