Maximum Satisfiability Solving Research Articles

From Single-Objective to Bi-Objective Maximum Satisfiability Solving

The declarative approach is key to efficiently finding optimal solutions to various types of NP-hard real-world combinatorial optimization problems. Most work on practical declarative solvers—ranging from classical integer programming to finite-domain constraint optimization and maximum satisfiability (MaxSAT)—has focused on optimization under a single objective; fewer advances have been made towards efficient declarative techniques for multi-objective optimization problems. Motivated by significant recent advances in practical solvers for MaxSAT, in this work we develop BiOptSat, an exact declarative approach for finding Pareto-optimal solutions to bi-objective optimization problems, with propositional logic as the underlying constraint language. BiOptSat can be viewed as an instantiation of the lexicographic method. The approach makes use of a single Boolean satisfiability solver that is incrementally employed throughout the entire search procedure, allowing for finding a single Pareto-optimal solution, finding one representative solution for each non-dominated point, and enumerating all Pareto-optimal solutions. We detail several algorithmic instantiations of BiOptSat, each building on recent algorithms proposed for single-objective MaxSAT. We empirically evaluate the instantiations compared to recently-proposed alternative approaches to multi-objective MaxSAT solving on several real-world domains from the literature, showing the practical benefits of our approach.

Unifying SAT-Based Approaches to Maximum Satisfiability Solving

Maximum satisfiability (MaxSAT), employing propositional logic as the declarative language of choice, has turned into a viable approach to solving NP-hard optimization problems arising from artificial intelligence and other real-world settings. A key contributing factor to the success of MaxSAT is the rise of increasingly effective exact solvers that are based on iterative calls to a Boolean satisfiability (SAT) solver. The three types of SAT-based MaxSAT solving approaches, each with its distinguishing features, implemented in current state-of-the-art MaxSAT solvers are the core-guided, the implicit hitting set (IHS), and the objective-bounding approaches. The objective-bounding approach is based on directly searching over the objective function range by iteratively querying a SAT solver if the MaxSAT instance at hand has a solution under different bounds on the objective. In contrast, both core-guided and IHS are so-called unsatisfiability-based approaches that employ a SAT solver as an unsatisfiable core extractor to determine sources of inconsistencies, but critically differ in how the found unsatisfiable cores are made use of towards finding a provably optimal solution. Furthermore, a variety of different algorithmic variants of the core-guided approach in particular have been proposed and implemented in solvers. It is well-acknowledged that each of the three approaches has its advantages and disadvantages, which is also witnessed by instance and problem-domain specific runtime performance differences (and at times similarities) of MaxSAT solvers implementing variants of the approaches. However, the questions of to what extent the approaches are fundamentally different and how the benefits of the individual methods could be combined in a single algorithmic approach are currently not fully understood. In this work, we approach these questions by developing UniMaxSAT, a general unifying algorithmic framework. Based on the recent notion of abstract cores, UniMaxSAT captures in general core-guided, IHS and objective-bounding computations. The framework offers a unified way of establishing quite generally the correctness of the current approaches. We illustrate this by formally showing that UniMaxSAT can simulate the computations of various algorithmic instantiations of the three types of MaxSAT solving approaches. Furthermore, UniMaxSAT can be instantiated in novel ways giving rise to new algorithmic variants of the approaches. We illustrate this aspect by developing a prototype implementation of an algorithmic variant for MaxSAT based on the framework.

Unifying SAT-Based Approaches to Maximum Satisfiability Solving

Maximum satisfiability (MaxSAT), employing propositional logic as the declarative language of choice, has turned into a viable approach to solving NP-hard optimization problems arising from artificial intelligence and other real-world settings. A key contributing factor to the success of MaxSAT is the rise of increasingly effective exact solvers that are based on iterative calls to a Boolean satisfiability (SAT) solver. The three types of SAT-based MaxSAT solving approaches, each with its distinguishing features, implemented in current state-of-the-art MaxSAT solvers are the core-guided, the implicit hitting set (IHS), and the objective-bounding approaches. The objective-bounding approach is based on directly searching over the objective function range by iteratively querying a SAT solver if the MaxSAT instance at hand has a solution under different bounds on the objective. In contrast, both core-guided and IHS are so-called unsatisfiability-based approaches that employ a SAT solver as an unsatisfiable core extractor to determine sources of inconsistencies, but critically differ in how the found unsatisfiable cores are made use of towards finding a provably optimal solution. Furthermore, a variety of different algorithmic variants of the core-guided approach in particular have been proposed and implemented in solvers. It is well-acknowledged that each of the three approaches has its advantages and disadvantages, which is also witnessed by instance and problem-domain specific runtime performance differences (and at times similarities) of MaxSAT solvers implementing variants of the approaches. However, the questions of to what extent the approaches are fundamentally different and how the benefits of the individual methods could be combined in a single algorithmic approach are currently not fully understood. In this work, we approach these questions by developing UniMaxSAT, a general unifying algorithmic framework. Based on the recent notion of abstract cores, UniMaxSAT captures in general core-guided, IHS and objective-bounding computations. The framework offers a unified way of establishing quite generally the correctness of the current approaches. We illustrate this by formally showing that UniMaxSAT can simulate the computations of various algorithmic instantiations of the three types of MaxSAT solving approaches. Furthermore, UniMaxSAT can be instantiated in novel ways giving rise to new algorithmic variants of the approaches. We illustrate this aspect by developing a prototype implementation of an algorithmic variant for MaxSAT based on the framework.

Efficient Learning of Interpretable Classification Rules

Machine learning has become omnipresent with applications in various safety-critical domains such as medical, law, and transportation. In these domains, high-stake decisions provided by machine learning necessitate researchers to design interpretable models, where the prediction is understandable to a human. In interpretable machine learning, rule-based classifiers are particularly effective in representing the decision boundary through a set of rules comprising input features. Examples of such classifiers include decision trees, decision lists, and decision sets. The interpretability of rule-based classifiers is in general related to the size of the rules, where smaller rules are considered more interpretable. To learn such a classifier, the brute-force direct approach is to consider an optimization problem that tries to learn the smallest classification rule that has close to maximum accuracy. This optimization problem is computationally intractable due to its combinatorial nature and thus, the problem is not scalable in large datasets. To this end, in this paper we study the triangular relationship among the accuracy, interpretability, and scalability of learning rule-based classifiers. The contribution of this paper is an interpretable learning framework IMLI, that is based on maximum satisfiability (MaxSAT) for synthesizing classification rules expressible in proposition logic. IMLI considers a joint objective function to optimize the accuracy and the interpretability of classification rules and learns an optimal rule by solving an appropriately designed MaxSAT query. Despite the progress of MaxSAT solving in the last decade, the straightforward MaxSAT-based solution cannot scale to practical classification datasets containing thousands to millions of samples. Therefore, we incorporate an efficient incremental learning technique inside the MaxSAT formulation by integrating mini-batch learning and iterative rule-learning. The resulting framework learns a classifier by iteratively covering the training data, wherein in each iteration, it solves a sequence of smaller MaxSAT queries corresponding to each mini-batch. In our experiments, IMLI achieves the best balance among prediction accuracy, interpretability, and scalability. For instance, IMLI attains a competitive prediction accuracy and interpretability w.r.t. existing interpretable classifiers and demonstrates impressive scalability on large datasets where both interpretable and non-interpretable classifiers fail. As an application, we deploy IMLI in learning popular interpretable classifiers such as decision lists and decision sets. The source code is available at https://github.com/meelgroup/mlic.

Learning from survey propagation: a neural network for MAX-E-3-SAT

Many natural optimization problems are NP-hard, which implies that they are probably hard to solve exactly in the worst-case. However, it suffices to get reasonably good solutions for all (or even most) instances in practice. This paper presents a new algorithm for computing approximate solutions in Θ(N) for the maximum exact 3-satisfiability (MAX-E-3-SAT) problem by using supervised learning methodology. This methodology allows us to create a learning algorithm able to fix Boolean variables by using local information obtained by the Survey Propagation algorithm. By performing an accurate analysis, on random conjunctive normal form instances of the MAX-E-3-SAT with several Boolean variables, we show that this new algorithm, avoiding any decimation strategy, can build assignments better than a random one, even if the convergence of the messages is not found. Although this algorithm is not competitive with state-of-the-art maximum satisfiability solvers, it can solve substantially larger and more complicated problems than it ever saw during training.

A MaxSAT-Based Framework for Group Testing

The success of MaxSAT (maximum satisfiability) solving in recent years has motivated researchers to apply MaxSAT solvers in diverse discrete combinatorial optimization problems. Group testing has been studied as a combinatorial optimization problem, where the goal is to find defective items among a set of items by performing sets of tests on items. In this paper, we propose a MaxSAT-based framework, called MGT, that solves group testing, in particular, the decoding phase of non-adaptive group testing. We extend this approach to the noisy variant of group testing, and propose a compact MaxSAT-based encoding that guarantees an optimal solution. Our extensive experimental results show that MGT can solve group testing instances of 10000 items with 3% defectivity, which no prior work can handle to the best of our knowledge. Furthermore, MGT has better accuracy than the LP-based approach. We also discover an interesting phase transition behavior in the runtime, which reveals the easy-hard-easy nature of group testing.

Efficient zonal diagnosis with maximum satisfiability

Model-based diagnosis (MBD) has been widely acknowledged as an effective diagnosis paradigm. However, for large scale circuits, it is difficult to find all cardinality-minimal diagnoses within a reasonable time. This paper proposes a novel method that takes a significant step in this direction. The idea is to divide a circuit into zones and compute the cardinality-minimal diagnoses by finding subset-minimal diagnoses with cardinality-minimal via a maximum satisfiability (MaxSAT) solver on an abstracted circuit that is composed of these zones instead of all components. We also propose a new propagate-extend method for extending the seed-TLDs to obtain all cardinality-minimal diagnoses efficiently. We implement our method with a state-of-the-art core-guided MaxSAT solver, and present evidence that it significantly improves the diagnosis efficiency on ISCAS-85 circuits. Our method outperforms SATbD, which was recently shown to outperform most complete MBD approaches using satisfiability (SAT).

An efficient method for reconfiguring power‐efficient VLSI array with maximum satisfiability

Techniques to determine subarrays when processing elements (PEs) of very large scale integration (VLSI) arrays become faulty have been investigated extensively. In this paper, we propose a maximum satisfiability (MaxSAT)‐based method for the reconfiguration of a two‐dimensional degradable VLSI array with faulty PEs. A MaxSAT model is developed such that the target array can be constructed utilizing the efficient MaxSAT solver. Using the proposed method, we are able to find a maximum logical array with the least number of long interconnects and minimizing the number of the long interconnects that will lead to less rerouting costs. Experimental results show that the proposed method is able to reduce most redundant interconnects for the host array compared to earlier works. © 2018 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

MSCG: Robust Core-Guided MaxSAT Solving

Maximum Satisfiability (MaxSAT) is a well-known optimization version of Propositional Satisfiability (SAT) that finds a wide range of relevant practical applications. This work describes and evaluates the Maximum Satisfiability using the Core-Guided approach solver (MSCG), which is a robust MaxSAT solver that participated in the MaxSAT Evaluation 2014. An experimental comparison with state-of-the-art MaxSAT solvers is presented.

On Using Incremental Encodings in Unsatisfiability-based MaxSAT Solving

In recent years, unsatisfiability-based algorithms have become prevalent as state of the art for solving industrial instances of Maximum Satisfiability (MaxSAT). These algorithms perform a succession of unsatisfiable SAT solver calls until an optimal solution is found. In several of these algorithms, cardinality or pseudo-Boolean constraints are extended between iterations. However, in most cases, the formula provided to the SAT solver in each iteration must be rebuilt and no knowledge is reused from one iteration to the next. This paper describes how to implement incremental unsatisfiability-based algorithms for MaxSAT. In particular, we detail and analyze our implementation of the MSU3 algorithm in the OPENWBO framework that performed remarkably well in the MaxSAT Evaluation of 2014. Furthermore, we also propose to extend incrementality to weighted MaxSAT through an incremental encoding of pseudo-Boolean constraints. The experimental results show that the performance of MaxSAT algorithms can be greatly improved by exploiting the learned information and maintaining the internal state of the SAT solver between iterations. Finally, the proposed incremental encodings of cardinality and pseudo-Boolean constraints are not exclusive for MaxSAT usage and can be used in other domains.

Cost-optimal constrained correlation clustering via weighted partial Maximum Satisfiability

Integration of the fields of constraint solving and data mining and machine learning has recently been identified within the AI community as an important research direction with high potential. This work contributes to this direction by providing a first study on the applicability of state-of-the-art Boolean optimization procedures to cost-optimal correlation clustering under constraints in a general similarity-based setting. We develop exact formulations of the correlation clustering task as Maximum Satisfiability (MaxSAT), the optimization version of the Boolean satisfiability (SAT) problem. For obtaining cost-optimal clusterings, we apply a state-of-the-art MaxSAT solver for solving the resulting MaxSAT instances optimally, resulting in cost-optimal clusterings. We experimentally evaluate the MaxSAT-based approaches to cost-optimal correlation clustering, both on the scalability of our method and the quality of the clusterings obtained. Furthermore, we show how the approach extends to constrained correlation clustering, where additional user knowledge is imposed as constraints on the optimal clusterings of interest. We show experimentally that added user knowledge allows clustering larger datasets, and at the same time tends to decrease the running time of our approach. We also investigate the effects of MaxSAT-level preprocessing, symmetry breaking, and the choice of the MaxSAT solver on the efficiency of the approach.

Deterministic Parallel MaxSAT Solving

Multicore processors are becoming the dominant platform in modern days. As a result, parallel Maximum Satisfiability (MaxSAT) solvers have been developed to exploit this new architecture. However, parallel MaxSAT solvers suffer from non-deterministic behavior, i.e. several runs of the same solver can lead to different solutions. This is a clear downside for applications that require solving the same problem instance more than once. This paper presents the first deterministic parallel MaxSAT solver that ensures reproducibility of results. Experimental results show that the performance of the new deterministic solver is comparable to the corresponding non-deterministic version.Another advantage of using a deterministic solver is the fact that one can easily observe the gains coming from different techniques, since the non-determinism is removed from the solver. For example, sharing learned clauses in parallel MaxSAT is expected to help to further prune the search space and boost the performance of a parallel solver. Yet, so far it has not been made clear which learned clauses should be shared among the different threads. By using the deterministic solver, we present a comparison showing that sharing learned clauses improves the overall performance of parallel MaxSAT solvers.

Improving linear search algorithms with model-based approaches for MaxSAT solving

Linear search algorithms have been shown to be particularly effective for solving partial Maximum Satisfiability (MaxSAT) problem instances. These algorithms start by adding a new relaxation variable to each soft clause and solving the resulting formula with a SAT solver. Whenever a Model is found, a new constraint on the relaxation variables is added such that Models with a greater or equal value are excluded. However, if the problem instance has a large number of relaxation variables, then adding a new constraint over these variables can lead to the exploration of a much larger search space. This article proposes new algorithms to improve the performance of linear search algorithms for MaxSAT by using Models found by the SAT solver to partition the relaxation variables. These algorithms add a new constraint on a subset of relaxation variables, thus intensifying the search on that subspace. In addition, the proposed algorithms are further enhanced by using incremental approaches where learned clauses are kept between calls to the SAT solver. Experimental results show that Model-based algorithms can outperform a classic linear search algorithm in several problem instances. Moreover, incremental approaches further improve the performance of linear search and Model-based algorithms for MaxSAT. Overall, Model-based algorithms are competitive with state-of-the-art MaxSAT solvers, and can improve their performance when solving problem instances with a large number of soft clauses.

Iterative and core-guided MaxSAT solving: A survey and assessment

Maximum Satisfiability (MaxSAT) is an optimization version of SAT, and many real world applications can be naturally encoded as such. Solving MaxSAT is an important problem from both a theoretical and a practical point of view. In recent years, there has been considerable interest in developing efficient algorithms and several families of algorithms have been proposed. This paper overviews recent approaches to handle MaxSAT and presents a survey of MaxSAT algorithms based on iteratively calling a SAT solver which are particularly effective to solve problems arising in industrial settings. First, classic algorithms based on iteratively calling a SAT solver and updating a bound are overviewed. Such algorithms are referred to as iterative MaxSAT algorithms. Then, more sophisticated algorithms that additionally take advantage of unsatisfiable cores are described, which are referred to as core-guided MaxSAT algorithms. Core-guided MaxSAT algorithms use the information provided by unsatisfiable cores to relax clauses on demand and to create simpler constraints. Finally, a comprehensive empirical study on non-random benchmarks is conducted, including not only the surveyed algorithms, but also other state-of-the-art MaxSAT solvers. The results indicate that (i) core-guided MaxSAT algorithms in general abort in less instances than classic solvers based on iteratively calling a SAT solver and that (ii) core-guided MaxSAT algorithms are fairly competitive compared to other approaches.