Josep Suy

A system for solving constraint satisfaction problems with SMT

SAT Modulo Theories (SMT) consists of deciding the satisfiability of a formula with respect to a decidable background theory, such as linear integer arithmetic, bit-vectors, etc, in first-order logic with equality. SMT has its roots in the field of verification. It is known that the SAT technology offers an interesting, efficient and scalable method for constraint solving, as many experimentations have shown. Although there already exist some results pointing out the adequacy of SMT techniques for constraint solving, there are no available tools to extensively explore such adequacy. In this paper we introduce a tool for translating FlatZinc (MiniZinc intermediate code) instances of constraint satisfaction problems to the standard SMT-LIB language. It can be used for deciding satisfiability as well as for optimization. The tool determines the required logic for solving each instance. The obtained results suggest that SMT can be effectively used to solve CSPs.

Solving constraint satisfaction problems with SAT modulo theories

Due to significant advances in SAT technology in the last years, its use for solving constraint satisfaction problems has been gaining wide acceptance. Solvers for satisfiability modulo theories (SMT) generalize SAT solving by adding the ability to handle arithmetic and other theories. Although there are results pointing out the adequacy of SMT solvers for solving CSPs, there are no available tools to extensively explore such adequacy. For this reason, in this paper we introduce a tool for translating FLATZINC (MINIZINC intermediate code) instances of CSPs to the standard SMT-LIB language. We provide extensive performance comparisons between state-of-the-art SMT solvers and most of the available FLATZINC solvers on standard FLATZINC problems. The obtained results suggest that state-of-the-art SMT solvers can be effectively used to solve CSPs.

Scheduling B2B Meetings

In this work we deal with the problem of scheduling meetings between research groups, companies and investors in a scientific and technological forum. We provide a CP formulation and a Pseudo-Boolean formulation of the problem, and empirically test the performance of different solving techniques, such as CP, lazy clause generation, SMT, and ILP, on industrial and crafted instances of the problem. The solutions obtained clearly improve expert handmade solutions with respect to the number of idle time slots and other quality parameters.

Solving weighted CSPs with meta-constraints by reformulation into satisfiability modulo theories

We introduce WSimply, a new framework for modelling and solving Weighted Constraint Satisfaction Problems (WCSP) using Satisfiability Modulo Theories (SMT) technology. In contrast to other well-known approaches designed for extensional representation of goods or no-goods, and with few declarative facilities, our approach aims to follow an intensional and declarative syntax style. In addition, our language has built-in support for some meta-constraints, such as priority and homogeneity, which allows the user to easily specify rich requirements on the desired solutions, such as preferences and fairness. We propose two alternative strategies for solving these WCSP instances using SMT. The first is the reformulation into Weighted SMT (WSMT) and the application of satisfiability test based algorithms from recent contributions in the Weighted Maximum Satisfiability field. The second one is the reformulation into an operation research-like style which involves an optimisation variable or objective function and the application of optimisation SMT solvers. We present experimental results of two well-known problems: the Nurse Rostering Problem (NRP) and a variant of the Balanced Academic Curriculum Problem (BACP), and provide some insights into the impact of the addition of meta-constraints on the quality of the solutions and the solving time.

Solving Intensional Weighted CSPs by Incremental Optimization with BDDs

We present a method for solving weighted Constraint Satisfaction Problems, based on translation into a Constraint Optimization Problem and iterative calls to an SMT solver, with successively tighter bounds of the objective function. The novelty of the method herewith described lies in representing the bound constraint as a shared Binary Decision Diagram, which in turn is translated into SAT. This offers two benefits: first, BDDs built for previous bounds can be used to build the BDDs for new (tighter) bounds, considerably reducing the BDD construction time; second, as a by-product, many clauses asserted to the solver in previous iterations can be reused.

MaxSAT-Based Scheduling of B2B Meetings

In this work we propose a MaxSAT formulation for the problem of scheduling business-to-business meetings. We identify some implied constraints and provide distinct encodings of the used cardinality constraints. The experimental results show that the proposed technique outperforms previous existing approaches on this problem.

Compact MDDs for Pseudo-Boolean Constraints with At-Most-One Relations in Resource-Constrained Scheduling Problems

Pseudo-Boolean (PB) constraints are usually encoded into Boolean clauses using compact Binary Decision Diagram (BDD) representations. Although these constraints appear in many problems, they are particularly useful for representing resource constraints in scheduling problems. Sometimes, the Boolean variables in the PB constraints have implicit at-most-one relations. In this work we introduce a way to take advantage of these implicit relations to obtain a compact Multi-Decision Diagram (MDD) representation for those PB constraints. We provide empirical evidence of the usefulness of this technique for some ResourceConstrained Project Scheduling Problem (RCPSP) variants, namely the Multi-Mode RCPSP (MRCPSP) and the RCPSP with Time-Dependent Resource Capacities and Requests (RCPSP/t). The size reduction of the representation of the PB constraints lets us decrease the number of Boolean variables in the encodings by one order of magnitude. We close/certify the optimum of many instances of these problems.

Solving the Multi-Mode Resource-Constrained Project Scheduling Problem with SMT

The Multi-Mode Resource-Constrained Project Scheduling Problem (MRCPSP) is a generalization of the well known Resource-Constrained Project Scheduling Problem (RCPSP). The most common exact approaches for solving this problem are based on branch-and-bound algorithms, mixed integer linear programming and Boolean satisfiability (SAT). In this paper, we present a new exact approach for solving this problem, using Satisfiability Modulo Theories (SMT). We provide two encodings into SMT and several reformulation and preprocessing techniques. The optimization algorithm that we propose uses an SMT solver as an oracle, and depending on its answer is able to update the encoding for the next optimization step. We report extensive performance experiments showing the utility of the proposed techniques and the good performance of our approach that allows us to close several open instances.

An exact approach for the prioritization process of industrial influents in wastewater systems

In wastewater systems, the efficiency of the treatment process is strongly related to the composition of its influent. When the treatment is overloaded (in volume and/or pollutants), its efficiency decreases and the effluent cannot attain the quality required by the receiving waters. This work considers the problem of mixing multiple wastewater streams, with multiple contaminants, into a single stream (the influent) on which various specifications are imposed. The problem has recently been solved by probabilistic methods that can achieve a nearly optimal solution. In this paper, an exact technique is proposed to find the optimal solution with a mixed-integer linear programming solver for the first time. The procedure is applied to a case study with different industrial effluents whose discharges will compose the influent to a treatment plant with constrained capacity (both in volume and pollutant loads). The optimal utility solution achieved describes the discharges that satisfy all constraints. This proposal constitutes an efficient way to manage treatment influents while reducing the computational time required by two orders of magnitude compared to probabilistic methods.

An Efficient SMT Approach to Solve MRCPSP/max Instances with Tight Constraints on Resources

The Multi-Mode Resource-Constrained Project Scheduling Problem with Minimum and Maximum Time Lags (MRCPSP/max) is a generalization of the well known Resource-Constrained Project Scheduling Problem. Recently, it has been shown that the benchmark datasets typically used in the literature can be easily solved by relaxing some resource constraints, which in many cases are dummy. In this work we propose new datasets with tighter resource limitations. We tackle them with an SMT encoding, where resource constraints are expressed as specialized pseudo-Boolean constraints and then translated into SAT. We provide empirical evidence that this approach is state-of-the-art for instances highly constrained by resources.