Complexity of chess domination problems

TL;DR

This paper investigates the computational complexity of domination problems involving rooks and queens on various polyominoes and polycubes, establishing NP-completeness results and providing computational tools and a related video game.

Contribution

It proves NP-completeness of maximum independent domination for non-attacking queens and rooks on 3D polycubes, and develops computational methods and a video game for these problems.

Findings

NP-completeness for 3D cases

New domination values for chessboards

A computational tool and a video game implementation

Abstract

We study different domination problems of attacking and non-attacking rooks and queens on polyominoes and polycubes of all dimensions. Our main result proves that maximum independent domination is NP-complete for non-attacking queens and for non-attacking rooks on polycubes of dimension three and higher. We also analyze these problems for polyominoes and convex polyominoes, conjecture the complexity classes, and provide a computer tool for investigation. We have also computed new values for classical queen domination problems on chessboards (square polyominoes). For our computations, we have translated the problem into an integer linear programming instance. Finally, using this computational implementation and the game engine Godot, we have developed a video game of minimum domination of queens and rooks on randomly generated polyominoes.