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 maximal domination is NP-complete for non-attacking queens and for non-attacking rooks on polycubes of dimension three and higher. We also analyse 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 minimal domination of queens and rooks on randomly generated polyominoes.

翻译：我们研究了不同维度多联骨牌和多联立方体上攻击与非攻击车和后的多种支配问题。主要结论证明：在三维及以上维度的多联立方体上，非攻击后与非攻击车的最大支配问题是NP完全的。我们还分析了多联骨牌和凸多联骨牌上的这些问题，推测了其复杂性类别，并开发了用于研究的计算机工具。针对棋盘（方形多联骨牌）上的经典后支配问题，我们计算得到了新的数值结果。在计算过程中，我们将问题转化为整数线性规划实例。最后，利用该计算实现方案与Godot游戏引擎，我们开发了一款基于随机生成多联骨牌的的最小化后与车支配视频游戏。