Partial MaxSAT Research Articles

Analysis of complex product design problem using an L-partition approach

In this paper, we analyze the complex product design problem using the model of the partial maximum satisfiability problem. In previous papers, estimates of the cardinality of L-complexes of polyhedrons of the SAT and the MAX SAT problems were obtained. In this paper, a relation of the cardinality of the L-complexes of the partial MAX SAT problem and the corresponding SAT problem is obtained. Using this result, it is possible to obtain theoretical estimates of the cardinality of the L-complex of the polyhedron of the complex product design problem on the basis of similar estimates for the SAT and the MAX SAT problems.

NuDist: An Efficient Local Search Algorithm for (Weighted) Partial MaxSAT

Abstract Maximum satisfiability (MaxSAT) is the optimization version of the satisfiability (SAT). Partial MaxSAT (PMS) generalizes SAT and MaxSAT by introducing hard and soft clauses, while Weighted PMS (WPMS) is the weighted version of PMS where each soft clause has a weight. These two problems have many important real-world applications. Local search is a popular method for solving (W)PMS. Recently, significant progress has been made in this direction by tailoring local search for (W)PMS, and a representative algorithm is the Dist algorithm. In this paper, we propose two ideas to improve Dist, including a clause-weighting scheme and a variable-selection heuristic. The resulting algorithm is called NuDist. Extensive experiments on PMS and WPMS benchmarks from the MaxSAT Evaluations (MSE) 2016 and 2017 show that NuDist significantly outperforms state-of-the-art local search solvers and performs better than state-of-the-art complete solvers including Open-WBO and WPM3 on MSE 2017 benchmarks. Also, empirical analyses confirm the effectiveness of the proposed ideas.

A comparative analysis and improvement of MaxSAT encodings for coalition structure generation under MC-nets

Abstract Coalition structure generation (CSG) is one of the main research issues in the use of coalitional games in multiagent systems and weighted partial MaxSAT (WPM) encodings, i.e. rule relation-based WPM (RWPM) and agent relation-based WPM (AWPM), which are efficient for solving the CSG problem. Existing studies show that AWPM surpasses RWPM since it achieves more compact encoding; it generates fewer variables and clauses than RWPM. However, in this paper, we focus on a special case in which the two encodings generate identical numbers of variables and clauses. Experiments show that RWPM surprisingly has a dominant advantage over AWPM, which aroused our interest. We exploit the deep-rooted reason and find that it is the redundancy when encoding transitive laws in RWPM that leads to this situation. Finally, we remove redundant clauses for transitive laws in RWPM and develop an improved RWPM with refined transitive laws to solve the CSG problem. Experiments demonstrate that refined encoding is more compact and efficient than previous WPM encodings.

Partial maximum satisfiability problem method combined with structure characteristics for diagnostic problems

Partial MaxSAT (PMS) is a generalized version of the maximum satisfiability problem (MaxSAT), which has important applications in many fields. At present, the PMS algorithm requires improvements in solving the industrial problems. Based on deep research on the PMS algorithm using random search, this paper presents a structure characteristics partial MaxSAT (SCPMS) method that combines structural feature exploration. First, according to the unit propagation rules and structural features of the problem, we gradually divide the PMS clause into two parts to construct a subproblem that can satisfy the problem due to the absence of the hard unit clause. In this paper, a random search guidance strategy is proposed. The new subproblem is reused to identify the hard-blocking variable in the hard unit clause of the original problem using unit propagation. Then, the corresponding soft-blocking variable is reversed with the feature of the clause to improve the efficiency of the random search. The experimental results show that in comparison with the two latest algorithms, DeciDist and DistUp, the proposed SCPMS greatly reduces the number of unsatisfied soft clauses in solving model-based diagnosis instances.

N-level Modulo-Based CNF encodings of Pseudo-Boolean constraints for MaxSAT

Many combinatorial problems in various fields can be translated to Maximum Satisfiability (MaxSAT) problems. Although the general problem is $\mathcal {N}\mathcal {P}$ -hard, more and more practical problems may be solved due to the significant effort which has been devoted to the development of efficient solvers. The art of constraints encoding is as important as the art of devising algorithms for MaxSAT. In this paper, we present several encoding methods of pseudo-Boolean constraints into Boolean satisfiability problems in Conjunctive Normal Form (CNF) formula, which are based on the idea of modular arithmetic and only generate auxiliary variables for each unique combination of weights. These techniques are efficient in encoding and solving MaxSAT problems. In particular, our solvers won the partial MaxSAT industrial category from 2010 through 2012 and ranked second in the 2017 main weighted track of the MaxSAT evaluation. We prove the correctness and the pseudo-polynomial space complexity of our encodings and also give a heuristics of the base selection for modular arithmetic. Our experimental results show that our encoding compactly encodes the constraints, and the obtained clauses are efficiently handled by a state-of-the-art SAT solver.

Improved WPM encoding for coalition structure generation under MC-nets

The Coalition Structure Generation (CSG) problem plays an important role in the domain of coalition games. Its goal is to create coalitions of agents so that the global welfare is maximized. To date, Weighted Partial MaxSAT (WPM) encoding has shown high efficiency in solving the CSG problem, which encodes a set of constraints into Boolean propositional logic and employs an off-the-shelf WPM solver to find out the optimal solution. However, in existing WPM encodings, a number of redundant encodings are asserted. This results in additional calculations and correspondingly incurs performance penalty. Against this background, this paper presents an Improved Rule Relation-based WPM (I-RWPM) encoding for the CSG problem, which is expressed by a set of weighted rules in a concise representation scheme called Marginal Contribution net (MC-net). In order to effectively reduce the constraints imposed on encodings, we first identify a subset of rules in an MC-net, referred as a set of freelance rules. We prove that solving the problem made up of all freelance rules can be achieved with a straightforward means without any extra encodings. Thus the set of rules requiring to be encoded is downsized. Next, we improve the encoding of transitive relations among rules. To be specific, compared with the existing rule relation-based encoding that generates transitive relations universally among all rules, I-RWPM only considers the transitivity among rules with particular relationship. In this way, the number of constraints to be encoded can be further decreased. Experiments suggest that I-RWPM significantly outperforms other WPM encodings for solving the same set of problem instances.

Modeling and solving staff scheduling with partial weighted maxSAT

Employee scheduling is a well known problem that appears in a wide range of different areas including health care, air lines, transportation services, and basically any organization that has to deal with workforces. In this paper we model a collection of challenging staff scheduling instances as a weighted partial Boolean maximum satisfiability (maxSAT) problem. Using our formulation we conduct a comparison of four different cardinality constraint encodings and analyze their applicability on this problem. Additionally, we measure the performance of two leading solvers from the maxSAT evaluation 2015 in a series of benchmark experiments and compare their results to state of the art solutions. In the process we also generate a number of challenging maxSAT instances that are publicly available and can be used as benchmarks for the development and verification of modern SAT solvers.

WPM3: An (in)complete algorithm for weighted partial MaxSAT

Maximum Satisfiability (MaxSAT) has been used to solve efficiently many combinatorial optimization problems. At recent editions of international MaxSAT Evaluation (MSE), the best performing solvers for real world (industrial) problems were those implementing SAT-based algorithms. These algorithms reformulate the MaxSAT optimization problem into a sequence of SAT decision problems where Pseudo-Boolean (PB) constraints may be introduced. In order to identify the most suitable PB constraints, some algorithms (core-guided) analyze the unsatisfiable cores retrieved from the previous SAT problems in the sequence while refining the lower bound. In this paper, we first conduct a comprehensive study on the complete core-guided algorithms Eva and OLL, that inspired the best performing solvers on industrial instances at MSE-2014. Despite of its apparently different foundations, we show how they are intimately related and identify how to improve them. In this sense, we present our complete core-guided algorithm WPM3. We show how to further exploit the analysis of unsatisfiable cores by being aware of their global structure, i.e., how the cores are interrelated. This is used to encode more efficient PB constraints and enables the algorithm to obtain assignments and refine also the upper bound. Therefore, WPM3 can also work as an incomplete algorithm. At MSE-2015, it showed a competitive performance on industrial instances. It got one out of three gold medals at the complete track and dominated at the incomplete track.

Reactive Dialectic Search Portfolios for MaxSAT

Metaheuristics have been developed to provide general purpose approaches for solving hard combinatorial problems. While these frameworks often serve as the starting point for the development of problem-specific search procedures, they very rarely work efficiently in their default state. We combine the ideas of reactive search, which adjusts key parameters during search, and algorithm configuration, which fine-tunes algorithm parameters for a given set of problem instances, for the automatic compilation of a portfolio of highly reactive dialectic search heuristics for MaxSAT. Even though the dialectic search metaheuristic knows nothing more about MaxSAT than how to evaluate the cost of a truth assignment, our automatically generated solver defines a new state of the art for random weighted partial MaxSAT instances. Moreover, when combined with an industrial MaxSAT solver, the self-assembled reactive portfolio was able to win four out of nine gold medals at the recent 2016 MaxSAT Evaluation on random, crafted, and industrial partial and weighted-partial MaxSAT instances.

An FPGA Solver for Partial MaxSAT Problems Based on Stochastic Local Search

In this paper, we propose an FPGA solver for partial maximum satisfiability (PMS) problems based on the Dist algorithm, which is one of the best performing stochastic local search algorithms for PMS problems. The Dist algorithm searches for a truth assignment for the variables that satisfies all of the hard clauses and as many soft clauses as possible by iteratively selecting a variable using a heuristic and flipping its truth value. During each iteration, new candidate variables for flipping are generated and existing ones may disappear. In our solver, the variables that may become new candidates for flipping are evaluated by parallel and pipeline processing, and then only the variables that actually become the candidates for flipping are extracted and gathered up in concurrent with the pipeline processing. The extraction process is not influenced by the number of the new candidates or their random generation, which minimizes the disturbance of the parallel and pipeline processing. Our FPGA solver can solve large PMS problems up to 7.74 times faster than running Dist on CPU.

CCEHC: An efficient local search algorithm for weighted partial maximum satisfiability

Weighted maximum satisfiability and (unweighted) partial maximum satisfiability (PMS) are two significant generalizations of maximum satisfiability (MAX-SAT), and weighted partial maximum satisfiability (WPMS) is the combination of the two, with more important applications in practice. Recently, great breakthroughs have been made on stochastic local search (SLS) for weighted MAX-SAT and PMS, resulting in several state-of-the-art SLS algorithms CCLS, Dist and DistUP. However, compared to the great progress of SLS on weighted MAX-SAT and PMS, the performance of SLS on WPMS lags far behind. In this paper, we present a new SLS algorithm named CCEHC for WPMS. CCEHC employs an extended framework of CCLS with a heuristic emphasizing hard clauses, called EHC. With strong accents on hard clauses, EHC has three components: a variable selection mechanism focusing on configuration checking based only on hard clauses, a weighting scheme for hard clauses, and a biased random walk component. Extensive experiments demonstrate that CCEHC significantly outperforms its state-of-the-art SLS competitors. Further experimental results on comparing CCEHC with a state-of-the-art complete solver show the effectiveness of CCEHC on a number of application WPMS instances, and indicate that CCEHC might be beneficial in practice. Also, empirical analyses confirm the effectiveness of each component underlying the EHC heuristic.

To be precise: regression aware debugging

Bounded model checking based debugging solutions search for mutations of program expressions that produce the expected output for a currently failing test. However, the current localization tools are not regression aware : they do not use information from the passing tests in their localization formula. On the other hand, the current repair tools attempt to guarantee regression freedom : when provided with a set of passing tests, they guarantee that none of these tests can break due to the suggested repair patch, thereby constructing a large repair formula. In this paper, we propose regression awareness as a means to improve the quality of localization and to scale repair. To enable regression awareness, we summarize the proof of correctness of each passing test by computing Craig Interpolants over a symbolic encoding of the passing execution, and use these summaries as additional soft constraints while synthesizing altered executions corresponding to failing tests. Intuitively, these additional constraints act as roadblocks, thereby discouraging executions that may damage the proof of a passing test. We use a partial MAXSAT solver to relax the proofs in a systematic way, and use a ranking function that penalizes mutations that damage the existing proofs. We have implemented our algorithms into a tool, TINTIN, that enables regression aware localization and repair. For localizations, our strategy is effective in extracting a superior ranking of suspicious locations: on a set of 52 different versions across 12 different programs spanning three benchmark suites, TINTIN achieves a saving of developer effort by almost 45% (in terms of the locations that must be examined by a developer to reach the ground-truth repair) in the worst case and 27% in the average case over existing techniques. For automated repairs, on our set of benchmarks, TINTIN achieves a 2.3X speedup over existing techniques without sacrificing much on the ranking of the repair patches: the ground-truth repair appears as the topmost suggestion in more than 70% of our benchmarks.

Optimal Partial-Order Plan Relaxation via MaxSAT

Partial-order plans (POPs) are attractive because of their least-commitment nature, which provides enhanced plan flexibility at execution time relative to sequential plans. Current research on automated plan generation focuses on producing sequential plans, despite the appeal of POPs. In this paper we examine POP generation by relaxing or modifying the action orderings of a sequential plan to optimize for plan criteria that promote flexibility. Our approach relies on a novel partial weighted MaxSAT encoding of a sequential plan that supports the minimization of deordering or reordering of actions. Using a similar technique, we further demonstrate how to remove redundant actions from the plan, and how to combine this criterion with the objective of maximizing a POP's flexibility. Our partial weighted MaxSAT encoding allows us to compute a POP from a sequential plan effectively. We compare the efficiency of our approach to previous methods for POP generation via sequential-plan relaxation. Our results show that while an existing heuristic approach consistently produces the optimal deordering of a sequential plan, our approach has greater flexibility when we consider reordering the actions in the plan while also providing a guarantee of optimality. We also investigate and confirm the accuracy of the standard flex metric typically used to predict the true flexibility of a POP as measured by the number of linearizations it represents.

New local search methods for partial MaxSAT

Maximum Satisfiability (MaxSAT) is the optimization version of the Satisfiability (SAT) problem. Partial Maximum Satisfiability (PMS) is a generalization of MaxSAT which involves hard and soft clauses and has important real world applications. Local search is a popular approach to solving SAT and MaxSAT and has witnessed great success in these two problems. However, unfortunately, local search algorithms for PMS do not benefit much from local search techniques for SAT and MaxSAT, mainly due to the fact that it contains both hard and soft clauses. This feature makes it more challenging to design efficient local search algorithms for PMS, which is likely the reason of the stagnation of this direction in more than one decade.In this paper, we propose a number of new ideas for local search for PMS, which mainly rely on the distinction between hard and soft clauses. The first three ideas, including weighting for hard clauses, separating hard and soft score, and a variable selection heuristic based on hard and soft score, are used to develop a local search algorithm for PMS called Dist. The fourth idea, which uses unit propagation with priority on hard unit clauses to generate the initial assignment, is used to improve Dist on industrial instances, leading to the DistUP algorithm.The effectiveness of our solvers and ideas is illustrated through experimental evaluations on all PMS benchmarks from the MaxSAT Evaluation 2014. According to our experimental results, Dist shows a significant improvement over previous local search solvers on all benchmarks. We also compare our solvers with state-of-the-art complete PMS solvers and a state-of-the-art portfolio solver, and the results show that our solvers have better performance in random and crafted instances but worse in industrial instances. The good performance of Dist has also been confirmed by the fact that Dist won all random and crafted categories of PMS and Weighted PMS in the incomplete solvers track of the MaxSAT Evaluation 2014.

On an MCS-based inconsistency measure

An important problem in knowledge-based systems is inconsistency handling. This problem has recently been attracting a lot of attention in AI community. In this paper, we tackle the problem of evaluating the amount of conflicts in knowledge bases, and provide a new fine grained inconsistency measure, denoted IMCC, based on maximal consistent sets (MCSes). The main idea consists in quantifying the inconsistency of a knowledge base by considering that all its consistent pieces of information are possible. Furthermore, we provide an epistemic interpretation of our inconsistency measure using the multimodal logic S5. Then, we show that IMCC satisfies several state-of-the-art postulates. Moreover, we provide an encoding in integer linear programming for computing our inconsistency measure, which is defined from the set of MCSes. We also propose a Partial Max-SAT encoding, which allows us to avoid the computation of the MCSes. Finally, we provide a comparison between IMCC and two related existing inconsistency measures.

Multi-language evaluation of exact solvers in graphical model discrete optimization

By representing the constraints and objective function in factorized form, graphical models can concisely define various NP-hard optimization problems. They are therefore extensively used in several areas of computer science and artificial intelligence. Graphical models can be deterministic or stochastic, optimize a sum or product of local functions, defining a joint cost or probability distribution. Simple transformations exist between these two types of models, but also with MaxSAT or linear programming. In this paper, we report on a large comparison of exact solvers which are all state-of-the-art for their own target language. These solvers are all evaluated on deterministic and probabilistic graphical models coming from the Probabilistic Inference Challenge 2011, the Computer Vision and Pattern Recognition OpenGM2 benchmark, the Weighted Partial MaxSAT Evaluation 2013, the MaxCSP 2008 Competition, the MiniZinc Challenge 2012 & 2013, and the CFLib (a library of Cost Function Networks). All 3026 instances are made publicly available in five different formats and seven formulations. To our knowledge, this is the first evaluation that encompasses such a large set of related NP-complete optimization frameworks, despite their tight connections. The results show that a small number of evaluated solvers are able to perform well on multiple areas. By exploiting the variability and complementarity of solver performances, we show that a simple portfolio approach can be very effective. This portfolio won the last UAI Evaluation 2014 (MAP task).

An Exact Algorithm Based on MaxSAT Reasoning for the Maximum Weight Clique Problem

Recently, MaxSAT reasoning is shown very effective in computing a tight upper bound for a Maximum Clique (MC) of a (unweighted) graph. In this paper, we apply MaxSAT reasoning to compute a tight upper bound for a Maximum Weight Clique (MWC) of a wighted graph. We first study three usual encodings of MWC into weighted partial MaxSAT dealing with hard clauses, which must be satisfied in all solutions, and soft clauses, which are weighted and can be falsified. The drawbacks of these encodings motivate us to propose an encoding of MWC into a special weighted partial MaxSAT formalism, called LW (Literal-Weighted) encoding and dedicated for upper bounding an MWC, in which both soft clauses and literals in soft clauses are weighted. An optimal solution of the LW MaxSAT instance gives an upper bound for an MWC, instead of an optimal solution for MWC. We then introduce two notions called the Top-k literal failed clause and the Top-k empty clause to extend classical MaxSAT reasoning techniques, as well as two sound transformation rules to transform an LW MaxSAT instance. Successive transformations of an LW MaxSAT instance driven by MaxSAT reasoning give a tight upper bound for the encoded MWC. The approach is implemented in a branch-and-bound algorithm called MWCLQ. Experimental evaluations on the broadly used DIMACS benchmark, BHOSLIB benchmark, random graphs and the benchmark from the winner determination problem show that our approach allows MWCLQ to reduce the search space significantly and to solve MWC instances effectively. Consequently, MWCLQ outperforms state-of-the-art exact algorithms on the vast majority of instances. Moreover, it is surprisingly effective in solving hard and dense instances.

On the performance of MaxSAT and MinSAT solvers on 2SAT-MaxOnes

We analyze and compare two solvers for Boolean optimization problems: WMaxSatz, a solver for Partial MaxSAT, and MinSatz, a solver for Partial MinSAT. Both MaxSAT and MinSAT are similar, but previous results indicate that when solving optimization problems using both solvers, the performance is quite different on some cases. For getting insights about the differences in the performance of the two solvers, we analyze their behaviour when solving 2SAT-MaxOnes problem instances, given that 2SAT-MaxOnes is probably the most simple, but NP-hard, optimization problem we can solve with them. The analysis is based first on the study of the bounds computed by both algorithms on some particular 2SAT-MaxOnes instances, characterized by the presence of certain particular structures. We find that the fraction of positive literals in the clauses is an important factor regarding the quality of the bounds computed by the algorithms. Then, we also study the importance of this factor on the typical case complexity of Random-p 2SAT-MaxOnes, a variant of the problem where instances are randomly generated with a probability p of having positive literals in the clauses. For the case p=0, the performance results indicate a clear advantage of MinSatz with respect to WMaxSatz, but as we consider positive values of p WMaxSatz starts to show a better performance, although at the same time the typical complexity of Random-p 2SAT-MaxOnes decreases as p increases. We also study the typical value of the bound computed by the two algorithms on these sets of instances, showing that the behaviour is consistent with our analysis of the bounds computed on the particular instances we studied first.

Mining Top-k motifs with a SAT-based framework

In this paper, we introduce a new problem, called Top-k SAT, that consists in enumerating the Top-k models of a propositional formula. A Top-k model is defined as a model with less than k models preferred to it with respect to a preference relation. We show that Top-k SAT generalizes two well-known problems: the Partial MAX-SAT problem and the problem of computing minimal models. Moreover, we propose a general algorithm for Top-k SAT. Then, we give an application of our declarative framework in data mining, namely, the problem of mining Top-k motifs in the transaction databases and in the sequences. In the case of mining sequence data, we introduce a new mining task by considering the sequences of itemsets. Thanks to the flexibility and to the declarative aspects of our SAT-based approach, an encoding of this task is obtained by a very slight modification of mining motifs in the sequences of items.

A Constraint Approach to Pivot-Based Bilingual Dictionary Induction

High-quality bilingual dictionaries are very useful, but such resources are rarely available for lower-density language pairs, especially for those that are closely related. Using a third language to link two other languages is a well-known solution and usually requires only two input bilingual dictionaries A-B and B-C to automatically induce the new one, A-C . This approach, however, has never been demonstrated to utilize the complete structures of the input bilingual dictionaries, and this is a key failing because the dropped meanings negatively influence the result. This article proposes a constraint approach to pivot-based dictionary induction where language A and C are closely related. We create constraints from language similarity and model the structures of the input dictionaries as a Boolean optimization problem, which is then formulated within the Weighted Partial Max-SAT framework, an extension of Boolean Satisfiability (SAT). All of the encoded CNF (Conjunctive Normal Form), the predominant input language of modern SAT/MAX-SAT solvers, formulas are evaluated by a solver to produce the target (output) bilingual dictionary. Moreover, we discuss alternative formalizations as a comparison study. We designed a tool that uses the Sat4j library as the default solver to implement our method and conducted an experiment in which the output bilingual dictionary achieved better quality than the baseline method.