Rectangular Spiral Galaxies are still hard

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