Computing Tighter Bounds on the $n$-Queens Constant via Newton's Method

TL;DR

This paper improves bounds on the $n$-queens problem exponent by applying Newton's method to convex optimization formulations, achieving tighter bounds than previous methods for large problem sizes.

Contribution

It introduces a Newton-based approach to solve convex optimization problems for the $n$-queens bounds, utilizing recent formulations and sharper bounds for improved results.

Findings

Achieved tighter bounds on the $n$-queens exponent

Demonstrated scalability of Newton's method for large instances

Enhanced previous bounds using improved formulations

Abstract

In recent work Simkin shows that bounds on an exponent occurring in the famous $n$-queens problem can be evaluated by solving convex optimization problems, allowing him to find bounds far tighter than previously known. In this note we use Simkin's formulation, a sharper bound developed by Knuth, and a Newton method that scales to large problem instances, to find even sharper bounds.