Generating Minimal Unsatisfiable SAT Instances from Strong Digraphs

Abstract

We present a model generator which generates SAT problems from digraphs. There are a few restrictions on the input digraphs. There must be no self-loops, and its vertices must be Boolean variables or labeled by distinct Boolean variables. We call such digraphs communication graphs. The model is pretty straightforward: if the communication graph contains the edges ($a, b$) and ($a, c$), and there is no other edge from $a$, then this is encoded by the clause: $\{\neg a, b, c\}$. The intuition is that $a$ can send a message to $b$ or $c$. We have to represent all cycles as well. If ($a, b, c, a$) is a cycle with the set of exit points $\{d, e\}$ in the input communication graph, then it is encoded by the clause: $\{\neg a,\neg b,\neg c, d, e\}$, The intuition is the following: if there is a message in the cycle, then it has to leave the cycle and we have to be sent to $d$ or $e$. We call this model as the weak model of communication graphs. We show that the weak model is a Black-and-White SAT problem if and only if the input is a strongly connected communication graph. We prove also that all clauses in such models are independent. From this we obtain that a weak model generated from a strong digraph is a minimal unsatisfiable SAT instance if we add to it the black and the white clauses, which are the only solutions of a Black-and-White SAT problems. Minimal unsatisfiable SAT instances are one of the hardest unsatisfiable clause sets, so they are interesting from the viewpoint of testing SAT solvers. There are some techniques which generates special minimal unsatisfiable SAT instances from digraphs, see the work of H. Abbasizanjani, and O. Kullmann, but there was no general solution before our work. Although, our solution is a general one, the generation of weak models is difficult because we have to find all cycles, including non-simple cycles. Therefore, we discuss how to create models of digraphs without cycle detection. Finally, we present some test results using state-of-the-art SAT solvers. It seems that these minimal unsatisfiable SAT instances are very difficult for them even with 20 variables.