Generalized Totalizer Encoding for Pseudo-Boolean Constraints

Abstract

Abstract. Pseudo-Boolean constraints, also known as 0-1 Integer Linear Constraints, are used to model many real-world problems. A common approach to solve these constraints is to encode them into a SAT formula. The runtime of the SAT solver on such formula is sensitive to the manner in which the given pseudo-Boolean constraints are encoded. In this paper, we propose generalized Totalizer encoding (GTE), which is an arc-consistency preserving extension of the Totalizer encoding to pseudo-Boolean constraints. Unlike some other encodings, the number of auxiliary variables required for GTE does not depend on the magnitudes of the coefficients. Instead, it depends on the number of distinct combinations of these coefficients. We show the superiority of GTE with respect to other encodings when large pseudo-Boolean constraints have low number of distinct coefficients. Our experimental results also show that GTE remains competitive even when the pseudo-Boolean constraints do not have this characteristic.

Approximation Strategies for Incomplete MaxSAT

“…We have implemented all the algorithms presented in this paper in Open-WBO-Inc. Open-WBO-Inc is built on top of Open-WBO [23] which uses Glucose [7] as the underlying SAT solver. We used Generalized Totalizer Encoding (GTE) [17] and incremental Totalizer encoding [22] to translate PB constraints and cardinality constraints into CNF, respectively.…”

“…These solvers use PB constraints to enforce convergence. While SAT4J [8] uses specialized data structures for PB constraints to avoid their conversion to CNF, other solvers such as QMaxSAT [18] convert the PB constraint into clauses using PB encodings [27,17,12]. Some MaxSAT solvers which are based on the implicit hitting set approach [11,25] maintain a lower and an upper bound on the values of the solution.…”

Core-Guided and Core-Boosted Search for CP

“…Similar constraints are encountered in many different domains, as such a lot of research has been put into developing efficient CNF encodings of them [46,11,27]. Even so, the PB constraint is arguably the main bottleneck of the overall performance of Lin-Search and we expect any further techniques that allow the use of simpler, and more compact (encodings) PB constraints to improve the overall performance of Lin-Search.…”

“…Specifically, such reformulations can decrease the size of the PB constraint P B = n i=1 w F (C i ) × r i < COST(F, τ ) that needs to be encoded during linear search. Depending on the specific encoding used, the number of clauses resulting from encoding P B into CNF depends either on the magnitudes of the weights of the soft clauses and the right-hand side [23] or on the number of unique sums that can be created from those weights [27]. The reformulation steps performed during the core-guided phase of a core-boosted algorithm can affect both of these factors.…”

Automatic Detection of At-Most-One and Exactly-One Relations for Improved SAT Encodings of Pseudo-Boolean Constraints

“…In this paper we focus on efficiently translating PB constraints to SAT within Savile Row, which produces a reformulated SAT model from an input constraint model in the Essence Prime language [26]. There exist several approaches for compactly encoding PB constraints to SAT based on different representations, such as Decision Diagrams [13,2], Sequential Weight Counters [17], Generalised Totalisers [19], and Polynomial Watchdog schemes [5].…”