Improved Lower Bounds for Queen’s Domination via an Exactly-Solvable Relaxation

The Queen's Domination problem, studied for over 160 years, poses the following question: What is the least number of queens that can be arranged on a m × n chessboard so that they either attack or occupy every cell? We propose a relaxation of the Queen's Domination problem and solve it exactly for rectangular chessboards. As a consequence, we improve on the best known lower bound for rectangular chessboards for one-eighth of the nontrivial cases. As another consequence, we generalize and provide a new interpretation of the best known lower bounds for Queen's Domination of square n × n chessboards for n ≡ {0, 1, 2} mod 4.Finally, we show some results and make some conjectures towards the goal of simplifying the long complicated proof for the best known lower bound for square boards when n ≡ 3 mod 4 (and n > 11). These simply stated conjectures may also be of independent interest.