On Optimization Modulo Theories, MaxSMT and Sorting Networks

Optimization Modulo Theories (OMT) is an extension of SMT which allows for finding models that optimize given objectives. (Partial weighted) MaxSMT --or equivalently OMT with Pseudo-Boolean objective functions, OMT+PB-- is a very-relevant strict subcase of OMT. We classify existing approaches for MaxSMT or OMT+PB in two groups: MaxSAT-based approaches exploit the efficiency of state-of-the-art MAXSAT solvers, but they are specific-purpose and not always applicable; OMT-based approaches are general-purpose, but they suffer from intrinsic inefficiencies on MaxSMT/OMT+PB problems. We identify a major source of such inefficiencies, and we address it by enhancing OMT by means of bidirectional sorting networks. We implemented this idea on top of the OptiMathSAT OMT solver. We run an extensive empirical evaluation on a variety of problems, comparing MaxSAT-based and OMT-based techniques, with and without sorting networks, implemented on top of OptiMathSAT and {\nu}Z. The results support the effectiveness of this idea, and provide interesting insights about the different approaches.Comment: 17 pages, submitted at Tacas 1

Speeding up the constraint-based method in difference logic

"The final publication is available at http://link.springer.com/chapter/10.1007%2F978-3-319-40970-2_18"Over the years the constraint-based method has been successfully applied to a wide range of problems in program analysis, from invariant generation to termination and non-termination proving. Quite often the semantics of the program under study as well as the properties to be generated belong to difference logic, i.e., the fragment of linear arithmetic where atoms are inequalities of the form u v = k. However, so far constraint-based techniques have not exploited this fact: in general, Farkas’ Lemma is used to produce the constraints over template unknowns, which leads to non-linear SMT problems. Based on classical results of graph theory, in this paper we propose new encodings for generating these constraints when program semantics and templates belong to difference logic. Thanks to this approach, instead of a heavyweight non-linear arithmetic solver, a much cheaper SMT solver for difference logic or linear integer arithmetic can be employed for solving the resulting constraints. We present encouraging experimental results that show the high impact of the proposed techniques on the performance of the VeryMax verification systemPeer ReviewedPostprint (author's final draft

Solving resource-constrained shceuling problems with exact methods

Scheduling problems mainly consist in finding an assignment of execution times (a schedule) to a set of activities of a project that optimizes an objective function. There are many constraints imposed over the activities that any schedule must satisfy. The most usual constraints establish precedence relations between activities, or limit the amount of some resources that the activities can consume. There are many scheduling problems in the literature that have been and are currently still being studied. A paradigmatic example is the Resource-Constraint Project Scheduling Problem (RCPSP). It consists in finding a start time for each one of the activities of a project, respecting pre-defined precedence relations between activities and without exceeding the capacity of a set of resources that the activities consume. The goal is to find a schedule with the minimum makespan (total execution time of the project). The RCPSP has many generalizations, one of which is the Multimode Resource-Constrained Project Scheduling Problem (MRCPSP). In this variation, each activity has several available execution modes that differ in the duration of the activity or the demand of resources. A solution for the MRCPSP determines the start times of the activities and also an execution mode for each one. These problems are NP-hard, and are known in the literature to be especially hard, with moderately small instances of 50 activities that are still open. There are many approaches to solving RCPSP and MRCPSP in the literature. They are often tackled with metaheuristics due to their high complexity, but there are also some exact approaches, including Mixed Integer Linear Programming (MILP), Branch-and-Bound algorithms or Boolean Satisfiability (SAT), which have shown to be competitive and in many cases even better than metaheuristics. One of the exact methods that is growing in use in the field of constrained optimization is SAT Modulo Theories (SMT). This thesis is the continuation of previous works carried out in the Logic and Programming (L ∧ P) group of Universitat de Girona, which used SMT to tackle RCPSP and MRCPSP. Excluding these, there have not been any other attempts to use SMT to solve the MRCPSP. SMT solvers (like other generic methods such as SAT or MILP) do not know which is the problem they are dealing with. It is the work of the modeler to provide a representation of the problem (i.e. an encoding) in the language that the solver admits. The main goal of this thesis is to use SMT to solve the Multimode Resource-Constraint Project Scheduling Problem. We focus on two already existing encodings for the MRCPSP, namely the time encoding and the task encoding. We use some existing preprocessing methods that contribute to the formulation of time and task, and present new preprocessings. Most of them are based on the idea of incompatibility between two activities, i.e., the impossibility that two activities run at the same time instant. These incompatibilities let us discharge some con- figurations of the solutions prior to encode the problem. Consequently, the use of preprocessings helps to reduce the size of the encodings in terms of variables and clauses. Another contribution of this work is the study of the time and task encodings and the differences that they present. We refine these encodings to provide more compact versions. Moreover, two new versions of these encodings are presented, which mainly differ in the codification of the constraints over the use of resources. One of them is based on Linear Integer Arithmetic expressions, and the other one in Pseudo-Boolean constraints and Integer Difference Logic. Another contribution of this work is the presentation of an ad-hoc optimization algorithm based on a linear search that mainly consists in three steps. First of all it simplifies the problem to efficiently ensure or discharge the feasibility of the instance, then it finds a first non-optimal solution by using a quick heuristic method, and finally it optimizes the problem making use of the knowledge acquired with the preprocessings to boost the search. We also present an initial work on a more intrusive approach consisting in modifying the internal heuristic of the SMT solver for the decision of literals. This work involves the study of a state-of-the-art implementation of an SMT solver, and its modification to include a framework to specify heuristics related with the encoding of the problem. We give some initial results on custom heuristics for the time and task encodings of the MRCPSP. Finally, we test our system with the benchmark sets of instances for the MRCPSP available in the literature, and compare our performance with a state-of-the-art exact solver for the MRCPSP. The results show that we are able to solve the major part of the benchmark sets. Moreover, we show to be competitive with the state-of-the-art solver of Vílim et. al. for the MRCPSP, being our system slower in solving the easiest benchmark instances, but outperforming the solver of Vílim et. al. in solving the hardest instance

A Comparison of SAT Encodings for Acyclicity of Directed Graphs

Many practical applications require synthesizing directed graphs that satisfy the acyclic constraint along with some side constraints. Several methods have been devised for encoding acyclicity of directed graphs into SAT, each of which is based on a cycle-detecting algorithm. The leaf-elimination encoding (LEE) repeatedly eliminates leaves from the graph, and judges the graph to be acyclic if the graph becomes empty at a certain time. The vertex-elimination encoding (VEE) exploits the property that the cyclicity of the resulting graph produced by the vertex-elimination operation entails the cyclicity of the original graph. While VEE is significantly smaller than the transitive-closure encoding for sparse graphs, it generates prohibitively large encodings for large dense graphs. This paper reports on a comparison study of four SAT encodings for acyclicity of directed graphs, namely, LEE using unary encoding for time variables (LEE-u), LEE using binary encoding for time variables (LEE-b), VEE, and a hybrid encoding which combines LEE-b and VEE. The results show that the hybrid encoding significantly outperforms the others