Jessica Davies

Solving MAXSAT by solving a sequence of simpler SAT instances

MAXSAT is an optimization version of Satisfiability aimed at finding a truth assignment that maximizes the satisfaction of the theory. The technique of solving a sequence of SAT decision problems has been quite successful for solving larger, more industrially focused MAXSAT instances, particularly when only a small number of clauses need to be falsified. The SAT decision problems, however, become more and more complicated as the minimal number of clauses that must be falsified increases. This can significantly degrade the performance of the approach. This technique also has more difficulty with the important generalization where each clause is given a weight: the weights generate SAT decision problems that are harder for SAT solvers to solve. In this paper we introduce a new MAXSAT algorithm that avoids these problems. Our algorithm also solves a sequence of SAT instances. However, these SAT instances are always simplifications of the initial MAXSAT formula, and thus are relatively easy for modern SAT solvers. This is accomplished by moving all of the arithmetic reasoning into a separate hitting set problem which can then be solved with techniques better suited to numeric reasoning, e.g., techniques from mathematical programming. As a result the performance of our algorithm is unaffected by the addition of clause weights. Our algorithm can, however, require solving more SAT instances than previous approaches. Nevertheless, the approach is simpler than previous methods and displays superior performance on some benchmarks.

Exploiting the power of MIP solvers in MAXSAT

maxsat is an optimization version of satisfiability. Since many practical problems involve optimization, there are a wide range of potential applications for effective maxsat solvers. In this paper we present an extensive empirical evaluation of a number of maxsat solvers. In addition to traditional maxsat solvers, we also evaluate the use of a state-of-the-art Mixed Integer Program (mip) solver, cplex, for solving maxsat. mip solvers are the most popular technology for solving optimization problems and are also theoretically more powerful than sat solvers. In fact, we show that cplex is quite effective on a range of maxsat instances. Based on these observations we extend a previously developed hybrid approach for solving maxsat, that utilizes both a sat solver and a mip solver. Our extensions aim to take better advantage of the power of the mip solver. The resulting improved hybrid solver is shown to be quite effective.

Preprocessing QBF

In this paper we investigate the use of preprocessing when solving Quantified Boolean Formulas (QBF). Many different problems can be efficiently encoded as QBF instances, and there has been a great deal of recent interest and progress in solving such instances efficiently. Ideas from QBF have also started to migrate to CSP with the exploration of Quantified CSPs which offer an intriguing increase in representational power over traditional CSPs. Here we show that QBF instances can be simplified using techniques related to those used for preprocessing SAT. These simplifications can be performed in polynomial time, and are used to preprocess the instance prior to invoking a worst case exponential algorithm to solve it. We develop a method for preprocessing QBF instances that is empirically very effective. That is, the preprocessed formulas can be solved significantly faster, even when we account for the time required to perform the preprocessing. Our method significantly improves the efficiency of a range of state-of-the-art QBF solvers. Furthermore, our method is able to completely solve some instances just by preprocessing, including some instances that to our knowledge have never been solved before by any QBF solver.

Postponing Optimization to Speed Up MAXSAT Solving

MAXSAT is an optimization version of sat that can represent a wide variety of important optimization problems. A recent approach for solving MAXSAT is to exploit both a SAT solver and a Mixed Integer Programming (MIP) solver in a hybrid approach. Each solver generates information used by the other solver in a series of iterations that terminates when an optimal solution is found. Empirical results indicate that a bottleneck in this process is the time required by the MIP solver, arising from the large number of times it is invoked. In this paper we present a modified approach that postpones the calls to the MIP solver. This involves substituting non-optimal solutions for the optimal ones computed by the mip solver, whenever possible. We describe the new approach and some different instantiations of it. We perform an extensive empirical evaluation comparing the performance of the resulting solvers with other state-of-the-art MAXSAT solvers. We show that the best performing versions of our approach advance the state-of-the-art in maxsat solving.

Exploiting resolution proofs to speed up LTL vacuity detection for BMC

When model-checking reports that a property holds on a model, vacuity detection increases user confidence in this result by checking that the property is satisfied in the intended way. While vacuity detection is effective, it is a relatively expensive technique requiring many additional model-checking runs. We address the problem of efficient vacuity detection for Bounded Model Checking (BMC) of linear temporal logic properties, presenting three partial vacuity detection methods based on the efficient analysis of the resolution proof produced by a successful BMC run. In particular, we define a characteristic of resolution proofs— peripherality—and prove that if a variable is a source of vacuity, then there exists a resolution proof in which this variable is peripheral. Our vacuity detection tool, VaqTree, uses these methods to detect vacuous variables, decreasing the total number of model-checking runs required to detect all sources of vacuity.

Computational protein design as an optimization problem

Proteins are chains of simple molecules called amino acids. The three-dimensional shape of a protein and its amino acid composition define its biological function. Over millions of years, living organisms have evolved a large catalog of proteins. By exploring the space of possible amino acid sequences, protein engineering aims at similarly designing tailored proteins with specific desirable properties. In Computational Protein Design (CPD), the challenge of identifying a protein that performs a given task is defined as the combinatorial optimization of a complex energy function over amino acid sequences. In this paper, we introduce the CPD problem and some of the main approaches that have been used by structural biologists to solve it, with an emphasis on the exact method embodied in the dead-end elimination/A? algorithm (DEE/A?). The CPD problem is a specific form of binary Cost Function Network (CFN, aka Weighted CSP). We show how DEE algorithms can be incorporated and suitably modified to be maintained during search, at reasonable computational cost. We then evaluate the efficiency of CFN algorithms as implemented in our solver toulbar2, on a set of real CPD instances built in collaboration with structural biologists. The CPD problem can be easily reduced to 0/1 Linear Programming, 0/1 Quadratic Programming, 0/1 Quadratic Optimization, Weighted Partial MaxSAT and Graphical Model optimization problems. We compare toulbar2 with these different approaches using a variety of solvers. We observe tremendous differences in the difficulty that each approach has on these instances. Overall, the CFN approach shows the best efficiency on these problems, improving by several orders of magnitude against the exact DEE/A? approach. The introduction of dead-end elimination before or during search allows to further improve these results.

Exploiting Resolution Proofs to Speed Up LTL Vacuity Detection for BMC

When model-checking reports that a property holds on a model, vacuity detection increases user confidence in this result by checking that the property is satisfied in the intended way. While vacuity detection is effective, it is a relatively expensive technique requiring many additional model-checking runs. We address the problem of efficient vacuity detection for Bounded Model Checking (BMC) of LTL properties, presenting three partial vacuity detection methods based on the efficient analysis of the resolution proof produced by a successful BMC run. In particular, we define a characteristic of resolution proofs - peripherality - and prove that if a variable is a source of vacuity, then there exists a resolution proof in which this variable is peripheral. Our vacuity detection tool, VaqTree, uses these methods to detect vacuous variables, decreasing the total number of model-checking runs required to detect all sources of vacuity.

Complexity of and algorithms for the manipulation of Borda, Nanson's and Baldwin's voting rules

We investigate manipulation of the Borda voting rule, as well as two elimination style voting rules, Nansons and Baldwins voting rules, which are based on Borda voting. We argue that these rules have a number of desirable computational properties. For unweighted Borda voting, we prove that it is NP-hard for a coalition of two manipulators to compute a manipulation. This resolves a long-standing open problem in the computational complexity of manipulating common voting rules. We prove that manipulation of Baldwins and Nansons rules is computationally more difficult than manipulation of Borda, as it is NP-hard for a single manipulator to compute a manipulation. In addition, for Baldwins and Nansons rules with weighted votes, we prove that it is NP-hard for a coalition of manipulators to compute a manipulation with a small number of candidates.Because of these NP-hardness results, we compute manipulations using heuristic algorithms that attempt to minimise the number of manipulators. We propose several new heuristic methods. Experiments show that these methods significantly outperform the previously best known heuristic method for the Borda rule. Our results suggest that, whilst computing a manipulation of the Borda rule is NP-hard, computational complexity may provide only a weak barrier against manipulation in practice. In contrast to the Borda rule, our experiments with Baldwins and Nansons rules demonstrate that both of them are often more difficult to manipulate in practice. These results suggest that elimination style voting rules deserve further study.

Using learnt clauses in MAXSAT

MAXSAT is an optimization version of SAT capable of expressing a variety of practical problems. MAXSAT solvers have been designed to take advantage of many of the successful techniques of SAT solvers. However, the most important technique of modern SAT solvers, clause learning, has not been utilized since learnt clauses cannot be soundly added to a MAXSAT theory. In this paper we present a new method that allows SAT clause learning to be exploited in a MAXSAT solver without losing soundness. We present techniques for learning clauses during a branch and bound (B&B) MAXSAT search, a process that is more complicated than standard clause learning. To exploit these learnt clauses we develop a connection between them and bounds that can be used during B&B. This connection involves formulating a hitting set problem and finding bounds on its optimal solution. We present some new techniques for generating useful hitting set bounds and also show how linear and integer programs can be exploited for this purpose, opening the door for a hybrid approach to solving MAXSAT.