The n-queens completion problem

Abstract

An n-queens configuration is a placement of n mutually non-attacking queens on an $$n\times n$$ n × n chessboard. The n-queens completion problem, introduced by Nauck in 1850, is to decide whether a given partial configuration can be completed to an n-queens configuration. In this paper, we study an extremal aspect of this question, namely: how small must a partial configuration be so that a completion is always possible? We show that any placement of at most n/60 mutually non-attacking queens can be completed. We also provide partial configurations of roughly n/4 queens that cannot be completed and formulate a number of interesting problems. Our proofs connect the queens problem to rainbow matchings in bipartite graphs and use probabilistic arguments together with linear programming duality.

Covering grids with multiplicity

“…In this section we introduce the linear programming method that will be used to prove our lower bounds. The use of linear programming in extremal combinatorics is well-established and has led to many results (see, for example, [GMCS22]), including the Clifton-Huang [CH20] lower bound on cov k (Γ) for Γ = {0, 1} d in the case when d is fixed and k tends to infinity. The standard template is as follows: assume without loss of generality that we are working with a minimization problem.…”