Path Puzzles: Discrete Tomography with a Path Constraint is Hard

We prove that path puzzles with complete row and column information–or equivalently, 2D orthogonal discrete tomography with Hamiltonicity constraint–are strongly NP-complete, ASP-complete, and #P-complete. Along the way, we newly establish ASP-completeness and #P-completeness for 3-Dimensional Matching and Numerical 3-Dimensional Matching.