Encoding Linear Constraints into SAT

Abstract

Linear integer constraints are one of the most important constraints in combinatorial problems since they are commonly found in many practical applications. Typically, encodings to Boolean satisfiability (SAT) format of conjunctive normal form perform poorly in problems with these constraints in comparison with SAT modulo theories (SMT), lazy clause generation (LCG) or mixed integer programming (MIP) solvers.In this paper we explore and categorize SAT encodings for linear integer constraints. We define new SAT encodings based on multi-valued decision diagrams, and sorting networks. We compare different SAT encodings of linear constraints and demonstrate where one may be preferable to another. We also compare SAT encodings against other solving methods and show they can be better than linear integer (MIP) solvers and sometimes better than LCG/SMT solvers on appropriate problems. Combining the new encoding with lazy decomposition, which during runtime only encodes constraints that are important to the solving process that occurs, gives the best option for many highly combinatorial problems involving linear constraints.

Portfolio approaches in constraint programming

“…There exists a lot of work proposing different techniques for encoding a CSP into a SAT problem [4,104,18,178,174]. A hybrid approach, called Lazy Clause Generation, is instead presented in [149].…”

“…The goal is no longer just finding a solution, but finding one that minimizes or maximizes the objective function. These kinds of problems are referred to as Constraint Optimization Problems (COPs) 4 .…”

“…Table 4.2: Solvers performances. 4 Note that the time metric is also called Penalized Average Runtime (PAR). Sometimes (e.g., in SAT competitions) PAR is replaced by PAR10, which penalizes the absence of answer with 10·T seconds instead of T seconds.…”

“…We have also filtered the initial dataset by removing, on one hand, the "easiest" instances (i.e., those for which the optimality was proven during the feature extraction) and, on the other, the "hardest" (i.e., those for which the features extraction has required more than T seconds). 4 The final dataset D on which we conducted our experiments thus consisted of 2670 instances.…”

“…We considered the aforementioned SATzilla and 3S approaches together with some OTS approaches (namely, IBk, J48, PART, RF, and SMO). Moreover, we have also implemented a generalization of 4 The time needed for extracting features was strongly dominated by the FlatZinc conversion.…”