pseudo-Boolean Solvers Research Articles

Pseudo-Boolean optimisation for RobinX sports timetabling

We report on the development of Reprobate, a tool for solving sports timetabling problems in a subset of the RobinX format. Our tool is based around a monolithic translation of a sports timetabling instance into a pseudo-Boolean (PB) optimisation problem; this instance can be solved using existing pseudo-Boolean solvers. Once the tool has found a feasible solution, it can improve it using a second encoding that alters only the home/away pattern of games. We entered our tool into the International Timetabling Competition 2021. While it was effective on many instances, it struggled to cope with schedules involving large break constraints. However, among instances for which it could initially find a feasible solution, the combination of use of a portfolio of solvers, a range of variations on the encoding and the aforementioned local improvement process yielded an average reduction in solution cost of 23%.

Cutting to the Core of Pseudo-Boolean Optimization: Combining Core-Guided Search with Cutting Planes Reasoning

Core-guided techniques have revolutionized Boolean satisfiability approaches to optimization problems (MaxSAT), but the process at the heart of these methods, strengthening bounds on solutions by repeatedly adding cardinality constraints, remains a bottleneck. Cardinality constraints require significant work to be re-encoded to SAT, and SAT solvers are notoriously weak at cardinality reasoning. In this work, we lift core-guided search to pseudo-Boolean (PB) solvers, which deal with more general PB optimization problems and operate natively with cardinality constraints. The cutting planes method used in such solvers allows us to derive stronger cardinality constraints, which yield better updates to solution bounds, and the increased efficiency of objective function reformulation also makes it feasible to switch repeatedly between lower-bounding and upper- bounding search. A thorough evaluation on applied and crafted benchmarks shows that our core-guided PB solver significantly improves on the state of the art in pseudo-Boolean optimization.

Learn to relax: Integrating 0-1 integer linear programming with pseudo-Boolean conflict-driven search

Conflict-driven pseudo-Boolean solvers optimize 0-1 integer linear programs by extending the conflict-driven clause learning (CDCL) paradigm from SAT solving. Though pseudo-Boolean solvers have the potential to be exponentially more efficient than CDCL solvers in theory, in practice they can sometimes get hopelessly stuck even when the linear programming (LP) relaxation is infeasible over the reals. Inspired by mixed integer programming (MIP), we address this problem by interleaving incremental LP solving with cut generation within the conflict-driven pseudo-Boolean search. This hybrid approach, which for the first time combines MIP techniques with full-blown conflict analysis operating directly on linear inequalities using the cutting planes method, significantly improves performance on a wide range of benchmarks, approaching a “best-of-both-worlds” scenario between SAT-style conflict-driven search and MIP-style branch-and-cut.

Employee Scheduling With SAT-Based Pseudo-Boolean Constraint Solving

The aim of this paper is practical: to show that, for at least one important real-world problem, modern SAT-based technology can beat the extremely mature branch-and-cut solving methods implemented in well-known state-of-the-art commercial solvers such as CPLEX or Gurobi . The problem of employee scheduling consists in assigning a work schedule to each of the employees of an organization, in such a way that demands are met, legal and contractual constraints are respected, and staff preferences are taken into account. This problem is typically handled by first modeling it as a 0–1 integer linear program (ILP). Our experimental setup considers as a case study the 0–1 ILPs obtained from the staff scheduling of a real-world car rental company, and carefully compares the performance of CPLEX and Gurobi with our own simple conflict-driven constraint-learning pseudo-Boolean solver.

1 A Novel Minimization Method for Sensor Deployment Via Heuristic 2-Sat Solution

The tasks of guard placement or sensor deployment in an art gallery, a museum or in the corridors of public and security buildings pose the same problem, which requires placing the guards or sensors so as to cover a specified set of nodes with a minimum number of sensors or guards, thereby reducing the overall cost of the system as well assist power consumption. Generally, minimization can be done using optimization techniques such as linear programming, but in case of sensor deployment or guard placement there is a need either to place or not to place the sensor or guard, and hence only Boolean or binary values are used. Therefore, in order to optimize such a problem, we use the special case of linear integer programming known as Boolean integer linear programming (0-1 ILP). Other algorithms like Pseudo-Boolean SAT Solvers can also be used for the minimization purpose. In this paper, we introduce these minimization algorithms for the sensor deployment problem. We also contribute a greed-based heuristic, which utilizes the fact that the pertinent propositional formulas have variables of purely un-complemented literals. This heuristic has much less computational cost compared to those of 0-1 ILP and the Pseudo-Boolean SAT Solvers.

A Novel Minimization Method for Sensor Deployment Via Heuristic 2-Sat Solution

The tasks of guard placement or sensor deployment in an art gallery, a museum or in the corridors of public and security buildings pose the same problem, which requires placing the guards or sensors so as to cover a specified set of nodes with a minimum number of sensors or guards, thereby reducing the overall cost of the system as well assist power consumption. Generally, minimization can be done using optimization techniques such as linear programming, but in case of sensor deployment or guard placement there is a need either to place or not to place the sensor or guard, and hence only Boolean or binary values are used. Therefore, in order to optimize such a problem, we use the special case of linear integer programming known as Boolean integer linear programming (0-1 ILP). Other algorithms like Pseudo-Boolean SAT Solvers can also be used for the minimization purpose. In this paper, we introduce these minimization algorithms for the sensor deployment problem. We also contribute a greed-based heuristic, which utilizes the fact that the pertinent propositional formulas have variables of purely un-complemented literals. This heuristic has much less computational cost compared to those of 0-1 ILP and the Pseudo-Boolean SAT Solvers.

A New Look at BDDs for Pseudo-Boolean Constraints

Pseudo-Boolean constraints are omnipresent in practical applications, and thus a significant effort has been devoted to the development of good SAT encoding techniques for them. Some of these encodings first construct a Binary Decision Diagram (BDD) for the constraint, and then encode the BDD into a propositional formula. These BDD-based approaches have some important advantages, such as not being dependent on the size of the coefficients, or being able to share the same BDD for representing many constraints. We first focus on the size of the resulting BDDs, which was considered to be an open problem in our research community. We report on previous work where it was proved that there are Pseudo-Boolean constraints for which no polynomial BDD exists. We also give an alternative and simpler proof assuming that NP is different from Co-NP. More interestingly, here we also show how to overcome the possible exponential blowup of BDDs by \emph{coefficient decomposition}. This allows us to give the first polynomial generalized arc-consistent ROBDD-based encoding for Pseudo-Boolean constraints. Finally, we focus on practical issues: we show how to efficiently construct such ROBDDs, how to encode them into SAT with only 2 clauses per node, and present experimental results that confirm that our approach is competitive with other encodings and state-of-the-art Pseudo-Boolean solvers.

PackUp: Tools for Package Upgradability Solving1

This paper presents PackUp (PACKage UPgradability with Boolean formulations) a framework for solving the the software package upgradability problem. Earlier versions of the framework (cudf2msu, cudf2pbo) participated in the 3rd MISC-live, an international competition organized by the European project MANCOOSI. The framework encodes the problem as a weighted partial MaxSAT formula and invokes a dedicated solver to solve the formula. The framework supports two types of solvers: weighted partial MaxSAT solvers and optimization pseudo-Boolean (OPB) solvers. The paper discusses the design of the framework and the specifics of the problem encoding.

Handling software upgradeability problems with MILP solvers

Upgradeability problems are a critical issue in modern operating systems. The problem consists in finding the "best" solution according to some criteria, to install, remove or upgrade packages in a given installation. This is a difficult problem: the complexity of the upgradeability problem is NP complete and modern OS contain a huge number of packages (often more than 20 000 packages in a Linux distribution). Moreover, several optimisation criteria have to be considered, e.g., stability, memory efficiency, network efficiency. In this paper we investigate the capabilities of MILP solvers to handle this problem. We show that MILP solvers are very efficient when the resolution is based on a linear combination of the criteria. Experiments done on real benchmarks show that the best MILP solvers outperform CP solvers and that they are significantly better than Pseudo Boolean solvers.

The Sat4j library, release 2.2

Sat4j is a mature, open source library of SAT-based solvers in Java. It provides a modular SAT solver architecture designed to work with generic constraints. Such architecture is used to provide SAT, MaxSat and pseudo-boolean and solvers for lightweight constraint programming. Those solvers have been evaluated regularly in the corresponding international competitive events. The library has been adopted by several academic softwares and the widely used Eclipse platform, which relies on a pseudo-boolean solver from Sat4j for its plugins dependencies management since June 2008.

The First Evaluation of Pseudo-Boolean Solvers (PB’05)

The rst evaluation of pseudo-Boolean solvers was organized as a subtrack of the SAT 2005 competition. The rst goal of this event is to take a snapshot of the current state of the art in the eld of pseudo-Boolean constraints. The second goal is to stimulate the research eorts in this eld and contribute to the creation of better technologies. This paper details the organization and the results of this event.

Translating Pseudo-Boolean Constraints into SAT

In this paper, we describe and evaluate three dieren t techniques for translating pseudoboolean constraints (linear constraints over boolean variables) into clauses that can be handled by a standard SAT-solver. We show that by applying a proper mix of translation techniques, a SAT-solver can perform on a par with the best existing native pseudo-boolean solvers. This is particularly valuable in those cases where the constraint problem of interest is naturally expressed as a SAT problem, except for a handful of constraints. Translating those constraints to get a pure clausal problem will take full advantage of the latest improvements in SAT research. A particularly interesting result of this work is the eciency of sorting networks to express pseudo-boolean constraints. Although tangential to this presentation, the result gives a suggestion as to how synthesis tools may be modied to produce arithmetic circuits more suitable for SAT based reasoning.