Properties of SAT Formulas Characterizing Convex Sets with Given Projections

Abstract

One of the most interesting and challenging problems in Discrete Tomography concerns the faithful reconstruction of an unknown finite discrete set from its horizontal and vertical projections. The computational complexity of this problem has been considered and solved in case of horizontal and vertical convex polyominoes, by coding the possible solutions through a 2-SAT formula. On the other hand, the problem is still open in case of (full) convex polyominoes. As a matter of fact, the previous polynomial-time reconstruction strategy does not naturally generalize to them. In particular, it has been observed that the convexity constraint on polyominoes involves, in general, a k-SAT formula $\varphi$, preventing, up to now, the polynomiality of the entire process, assuming that $P \ne NP$. Our studies focus on the clauses of $\varphi$. We show that they can be reduced to 2-SAT or 3-SAT only and that a subset of the variables involved in the reconstruction may appear in the 3-SAT clauses of $\varphi$, thus detecting some situations that lead to a polynomial time reconstruction. Some examples of situations where 3-SAT formulas arise are also provided.