Towards Robust CNF Encodings of Cardinality Constraints

Abstract

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.

Haplotype Inference with Boolean Constraint Solving: An Overview

“…In current implementations, auxiliary variables are not handled differently by SAT solvers. However, the special handling of these variables may be considered in the future, as it has been recently shown that not branching on these variables, as well as not learning constraints with these variables, leads in general to more robust performances [15]. Besides the basic model outlined above, SAT-based haplotyping requires the inclusion of a number of effective techniques, including lower bounds and identification of symmetries [11] (see Section 4.3).…”

A Lower Bound on CNF Encodings of the At-Most-One Constraint

“…Encodings using auxiliary variables are mainly useful for constraints on a large number of input variables, when the full prime representation is too large. Moreover, the experimental results in [19] suggest that a SAT solver can be modified to minimize the disadvantage of introducing auxiliary variables. Another experimental evaluation of various cardinality constraints and their encodings appears in [15].…”

Exploiting Short Supports for Improved Encoding of Arbitrary Constraints into SAT

“…We add to the important and growing literature on modelling of constraints in SAT [6]. Most study has been devoted to constraints such as linear constraints including the special case of cardinality constraints [1,2,8,24].…”