Generalized Totalizer Encoding for Pseudo-Boolean Constraints

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.Comment: 10 pages, 2 figures, 2 tables. To be published in 21st International Conference on Principles and Practice of Constraint Programming 201

Approximate-At-Most-k Encoding of SAT for Soft Constraints

In the field of Boolean satisfiability problems (SAT), at-most-k constraints, which suppress the number of true target variables at most k, are often used to describe objective problems. At-most-k constraints are used not only for absolutely necessary constraints (hard constraints) but also for challenging constraints (soft constraints) to search for better solutions. To encode at-most-k constraints into Boolean expressions, there is a problem that the number of Boolean expressions basically increases exponentially with the number of target variables, so at-most-k often has difficulties for a large number of variables. To solve this problem, this paper proposes a new encoding method of at-most-k constraints, called approximate-at-most-k, which has totally fewer Boolean expressions than conventional methods on the one hand. On the other hand, it has lost completeness, i.e., some Boolean value assignments that satisfy the original at-most-k are not allowed with approximate-at-most-k; hence, it is called approximate. Without completeness, we still have potential benefits by using them only as soft constraints. For example, approximate-at-most-16 out of 32 variables requires only 15% of a conventional at-most-k on the literal number and covers 44% of the solution space. Thus, approximate-at-most-k can become an alternative encoding method for at-most-k, especially as soft constraints.Comment: 9 pages, 6 figures, 14th Pragmatics of SAT international workshop (PoS2023): accepte