Path Puzzles: Discrete Tomography with a Path Constraint is Hard

Path Puzzles are a type of logic puzzle introduced in Roderick Kimball’s 2013 book [5]. A puzzle consists of a (rectangular) grid of cells with two exits (or “doors”) on the boundary and numerical constraints on some subset of the rows and columns. A solution consists of a single non-intersecting path which starts and ends at two boundary doors and which passes through a number of cells in each constrained row and column equal to the given numerical clue. Figure 1 shows some example path puzzles and Figure 4 shows their (unique) solutions. Many variations of path puzzles are given in [5] and elsewhere, for example using non-rectangular grids, grid-internal constraints, and additional candidate doors, but these generalizations make the problem only harder. A path puzzle can be seen as 2-dimensional discrete tomography [3] problem with partial information (not all row and column sums) and an additional Hamiltonicity constraint on the output image. Vanilla 2-dimensional discrete tomography is known to have efficient (polynomialtime) algorithms [3], though it becomes hard under certain connectivity constraints on the output image [2]. Our results. Unlike 2-dimensional discrete tomography, we show that path puzzles (with partial information and the added Hamiltonicity constraint) are in fact NP-complete. In fact, we prove the stronger results that path puzzles are Another Solution Problem (ASP) hard and (to count solutions) #P-