Towards Robust CNF Encodings of Cardinality Constraints

Abstract

Motivated by the performance improvements made to SAT solvers in recent years, a number of different encodings of constraints into SAT have been proposed. Concrete examples are the different SAT encodings for ≤ 1 (x1, . . . , xn) constraints. The most widely used encoding is known as the pairwise encoding, which is quadratic in the number of variables in the constraint. Alternative encodings are in general linear, and require using additional auxiliary variables. In most settings, the pairwise encoding performs acceptably well, but can require unacceptably large Boolean formulas. In contrast, linear encodings yield much smaller Boolean formulas, but in practice SAT solvers often perform unpredictably. This lack of predictability is mostly due to the large number of auxiliary variables that need to be added to the resulting Boolean formula. This paper studies one specific encoding for ≤ 1 (x1, . . . , xn) constraints, and shows how a state-of-the-art SAT solver can be adapted to overcome the problem of adding additional auxiliary variables. Moreover, the paper shows that a SAT solver may essentially ignore the existence of auxiliary variables. Experimental results indicate that the modified SAT solver becomes significantly more robust on SAT encodings involving ≤ 1 (x1, . . . , xn) constraints.