Rectangular Spiral Galaxies are Still Hard

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 $180^\circ$ rotationally symmetric about its center. We show that this puzzle is NP-complete even if the polyominoes are restricted to be (a) rectangles of arbitrary size or (b) 1$\times$1, 1$\times$3, and 3$\times$1 rectangles. The proof for the latter variant also implies NP-completeness of finding a non-crossing matching in modified grid graphs where edges connect vertices of distance $2$. Moreover, we prove NP-completeness of the design problem of minimizing the number of centers such that there exist a set of Spiral Galaxies that exactly cover a given shape.