Complexity of chess domination problems

TL;DR

The paper addresses the complexity of domination and maximum independent domination problems for rooks and queens on polyominoes and polycubes across dimensions. It introduces divisible gadget-based reductions from planar 3-SAT variants to prove NP-completeness for $d\ge3$, for both minimum domination and maximum independent domination, and it extends the analysis to polyominoes and convex polyominoes with conjectures. A unified ILP formulation and an accompanying solver are developed, enabling computation of new domination numbers on $n\times n$ boards up to $n=31$ and supporting a game-based exploration platform. Collectively, the work advances understanding of domination problems in higher dimensions, provides practical computational tools, and outlines open questions and conjectures for future research.

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.

Figures (23)

• Figure 1: Possible movement (in green) of a rook and a queen centered in a $3\times 3\times 3$ cube. For the queen, a $5\times 5$ level is put at the top of the cube. 3D models corresponding to these structures can be found at https://skfb.ly/oz8tJ and https://skfb.ly/oz8tn, respectively.
• Figure 2: An instance of $\mathrm{P3SAT}_{3}$ with three variables, $x_1$, $x_2$ and $x_3$, and four clauses, $c_1=x_1\lor x_2\lor x_3$, $c_2=\overline x_1\lor \overline x_2$, $c_3=\overline x_2\lor \overline x_3$ and $c_4=x_1\lor \overline x_3$. There are three solutions: $(x_1,x_2,x_3) = (1,0,1),\ (1,0,0),\ (0,1,0)$
• Figure 3: Left: an instance of a polyomino with attacking dominating set of 2 queens and non-attacking dominating set of 3 queens; right: an instance of a polycube with attacking dominating set of 2 rooks and non-attacking of 3 rooks.
• Figure 4: Variable gadget with rooks; when true, 2 additional rooks go on the dark cubes (T), and when false, 2 go on the light red cubes (F). A 3D model corresponding to this structure can be found at https://skfb.ly/oz8tZ. The general orientation is given next to it.
• Figure 5: The connection gadget for rooks; values are transmitted along the T and F cubes for true and false, respectively. There are some cubes not shown on the image. (A): the left has a T cube not shown; right has an F cube not shown; (B): two T cubes not shown behind and under the F cube of the connection gadget on the variable gadget

Theorems & Definitions (61)

• Definition
• Definition : Rook attacking powers
• Definition : Queen attacking powers alpert2021art
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition : Independent rook domination
• Definition : Independent queen domination
• Definition : $\mathrm{P3SAT}_{3}$ Cerioli08