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

A Propositional CONEstrip Algorithm

We present a variant of the CONEstrip algorithm for checking whether the origin lies in a finitely generated convex cone that can be open, closed, or neither. This variant is designed to deal efficiently with problems where the rays defining the cone are specified as linear combinations of propositional sentences. The variant differs from the original algorithm in that we apply row generation techniques. The generator problem is WPMaxSAT, an optimization variant of SAT; both can be solved with specialized solvers or integer linear programming techniques. We additionally show how optimization problems over the cone can be solved by using our propositional CONEstrip algorithm as a preprocessor. The algorithm is designed to support consistency and inference computations within the theory of sets of desirable gambles. We also make a link to similar computations in probabilistic logic, conditional probability assessments, and imprecise probability theory

Reducing chaos in SAT-like search: finding solutions close to a given one

Motivated by our own industrial users, we attack the following challenge that is crucial in many practical planning, scheduling or timetabling applications. Assume that a solver has found a solution for a given hard problem and, due to unforeseen circumstances (e.g., rescheduling), or after an analysis by a committee, a few more constraints have to be added and the solver has to be re-run. Then it is almost always important that the new solution is “close” to the original one. The activity-based variable selection heuristics used by SAT solvers make search chaotic, i.e., extremely sensitive to the initial conditions. Therefore, re-running with just one additional clause added at the end of the input usually gives a completely different solution. We show that naive approaches for finding close solutions do not work at all, and that solving the Boolean optimization problem is far too inefficient: to find a reasonably close solution, state-of-the-art tools typically require much more time than was needed to solve the original problem. Here we propose the first (to our knowledge) approach that obtains close solutions quickly. In fact, it typically finds the optimal (i.e., closest) solution in only 25% of the time the solver took in solving the original problem. Our approach requires no deep theoretical or conceptual innovations. Still, it is non-trivial to come up with and will certainly be valuable for researchers and practitioners facing the same problem.Postprint (published version

Search Pruning Techniques in SAT-Based Branch-and-Bound Algorithms for the Binate Covering Problem

Abstract—Covering problems are widely used as a modeling tool in electronic design automation. Recent years have seen dramatic improvements in algorithms for the unate/binate cov-ering problem (UCP/BCP). Despite these improvements, BCP is a well-known computationally hard problem with many existing real-world instances that currently are hard or even impossible to solve. In this paper we apply search pruning techniques from the Boolean satisfiability domain to branch-and-bound algorithms for BCP. Furthermore, we generalize these techniques, in particular the ability to infer and record new constraints from conflicts and the ability to backtrack nonchronologically, to situations where the branch-and-bound BCP algorithm backtracks due to bounding conditions. Experimental results, obtained on representative real-world in-stances of the UCP/BCP, indicate that the proposed techniques are effective and can provide significant performance gains for specific classes of instances. Index Terms—Backtrack search, binate covering problem, branch-and-bound, nonchronological backtracking, propositional satisfiability. I

On Weakening Strategies for PB Solvers

International audienceCurrent pseudo-Boolean solvers implement different variants of the cutting planes proof system to infer new constraints during conflict analysis. One of these variants is generalized resolution, which allows to infer strong constraints, but suffers from the growth of coefficients it generates while combining pseudo-Boolean constraints. Another variant consists in using weakening and division, which is more efficient in practice but may infer weaker constraints. In both cases, weakening is mandatory to derive conflicting constraints. However, its impact on the performance of pseudo-Boolean solvers has not been assessed so far. In this paper, new application strategies for this rule are studied, aiming to infer strong constraints with small coefficients. We implemented them in Sat4j and observed that each of them improves the runtime of the solver. While none of them performs better than the others on all benchmarks, applying weakening on the conflict side has surprising good performance, whereas applying partial weakening and division on both the conflict and the reason sides provides the best results overall