Rectangular Spiral Galaxies are Still Hard

TL;DR

This paper proves the computational complexity of a puzzle called Spiral Galaxies, showing it is NP-complete and related problems are also hard, even under restrictive conditions.

Contribution

It establishes the NP-completeness, ASP-completeness, and #P-completeness of the Spiral Galaxies puzzle and related problems, including variants with rectangular polyominoes.

Findings

Proves NP-completeness of the Spiral Galaxies puzzle.

Shows the problem remains hard even with rectangular constraints.

Establishes complexity results for related graph matching and design problems.

Abstract

Spiral Galaxies is a pencil-and-paper puzzle played on a grid of unit squares: given a set of points called centers, the goal is to partition the grid into polyominoes such that each polyomino contains exactly one center and is 180{\deg} rotationally symmetric about its center. We show that this puzzle is NP-complete, ASP-complete, and #P-complete even if (a) all solutions to the puzzle have rectangles for polyominoes; or (b) the polyominoes are required to be rectangles and all solutions to the puzzle have just 1$\times$1, 1$\times$3, and 3$\times$1 rectangles. The proof for the latter variant also implies NP/ASP/#P-completeness of finding a noncrossing perfect matching in distance-2 grid graphs where edges connect vertices of Euclidean distance 2. Moreover, we prove NP-completeness of the design problem of minimizing the number of centers such that there exists a set of galaxies that exactly cover a given shape