Unifying SAT-Based Approaches to Maximum Satisfiability Solving

Maximum Satisfiability (MaxSAT), the optimisation extension of the well-known Boolean Satisfiability (SAT) problem, is a competitive approach for solving NP-hard problems encountered in various artificial intelligence and industrial domains. Due to its computational complexity, there is an inherent tradeoff between scalability and guarantee on solution quality in MaxSAT solving. Limitations on available computational resources in many practical applications motivate the development of complete any-time MaxSAT solvers, i.e. algorithms that compute optimal solutions while providing intermediate results. In this work, we propose core-boosted linear search, a generic search-strategy that combines two central approaches in modern MaxSAT solving, namely linear and core-guided algorithms. Our experimental evaluation on a prototype combining reimplementations of two state-of-the-art MaxSAT solvers, PMRES as the core-guided approach and LinSBPS as the linear algorithm, demonstrates that our core-boosted linear algorithm often outperforms its individual components and shows competitive and, on many domains, superior results when compared to other state-of-the-art solvers for incomplete MaxSAT solving.

Maximum Satisfiability (MaxSAT) is a general model for formulating combinatorial optimization problems. MaxSAT formulas encoded from different domains have different features, yet most MaxSAT solvers are designed for general formulas. This work considers an important subclass of MaxSAT, named as Pure MaxSAT, which characterizes a wide range of combinatorial optimization problems particularly subset problems. We design a novel local search method for Pure MaxSAT, which combines the idea of linear search and local search and is dubbed as linear local search. Our algorithm LinearLS significantly outperforms state of the art MaxSAT solvers on Pure MaxSAT instances, including instances from MaxSAT Evaluations and those encoded from three famous NP hard combinatorial optimization problems. Moreover, LinearLS outperforms state of the art algorithms for each tested combinatorial optimization problem on the popular benchmarks.

Maximum satisfiability (MaxSAT) is a viable approach to solving NP-hard optimization problems. In the realm of core-guided MaxSAT solving - one of the most effective MaxSAT solving paradigms today - algorithmic variants employing so-called soft cardinality constraints have proven very effective. In this work, we propose to combine weight-aware core extraction (WCE) - a recently proposed approach that enables relaxing multiple cores instead of a single one during iterations of core-guided search - with a novel form of structure sharing in the cardinality-based core relaxation steps performed in core-guided MaxSAT solvers. In particular, the proposed form of structure sharing is enabled by WCE, which has so-far not been widely integrated to MaxSAT solvers, and allows for introducing fewer variables and clauses during the MaxSAT solving process. Our results show that the proposed techniques allow for avoiding potential overheads in the context of soft cardinality constraint based core-guided MaxSAT solving both in theory and in practice. In particular, the combination of WCE and structure sharing improves the runtime performance of a state-of-the-art core-guided MaxSAT solver implementing the central OLL algorithm.

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.

Maximum satisfiability (MaxSAT) is an optimization version of SAT that is solved by finding an optimal truth assignment instead of just a satisfying one. In MaxSAT the objective function to be optimized is specified by a set of weighted soft clauses: the objective value of a truth assignment is the sum of the weights of the soft clauses it satisfies. In addition, the MaxSAT problem can have hard clauses that the truth assignment must satisfy. Many optimization problems can be naturally encoded into MaxSAT and this, along with significant performance improvements in MaxSAT solvers, has led to MaxSAT being used in a number of different application areas. This chapter provides a detailed overview of the approaches to MaxSAT solving that have in recent years been most successful in solving real-world optimization problems. Further recent developments in MaxSAT research are also overviewed, including encodings, applications, preprocessing, incomplete solving, algorithm portfolios, partitioning-based solving, and parallel solving.

Boolean Satisfiability (SAT), the first problem proven to be NP-complete, is intractable on digital computers based on the von Neumann architecture. An efficient SAT solver can benefit many applications such as artificial intelligence, circuit design, and functional verification. Recently, a SAT solver approach based on a deterministic, continuous-time dynamical system (CTDS) was introduced [1]. This approach shows polynomial analog time-complexity on even the hardest k-SAT (k ≥ 3) problem instances, but at an energy cost dependent on exponentially growing auxiliary variables. This paper reports a novel analog hardware SAT solver, AC-SAT, implementing the CTDS via incorporating novel, analog circuit design ideas. AC-SAT is intended to be used as a co-processor and is programmable for handling different problem specifications. Furthermore, with its modular design, AC-SAT can be readily extended to solve larger size problems. SPICE simulation results show that AC-SAT can indeed solve the SAT problems, and it has speedup factors of ~10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> on even the hardest 3-SAT problems, when compared with a state-of-the-art SAT solver on digital computers.

Chapter 2 Satisfiability Solvers

The study of clause redundancy in Boolean satisfiability (SAT) has proven significant in various terms, from fundamental insights into preprocessing and inprocessing to the development of practical proof checkers and new types of strong proof systems. We study liftings of the recently-proposed notion of propagation redundancy—based on a semantic implication relationship between formulas—in the context of maximum satisfiability (MaxSAT), where of interest are reasoning techniques that preserve optimal cost (in contrast to preserving satisfiability in the realm of SAT). We establish that the strongest MaxSAT-lifting of propagation redundancy allows for changing in a controlled way the set of minimal correction sets in MaxSAT. This ability is key in succinctly expressing MaxSAT reasoning techniques and allows for obtaining correctness proofs in a uniform way for MaxSAT reasoning techniques very generally. Bridging theory to practice, we also provide a new MaxSAT preprocessor incorporating such extended techniques, and show through experiments its wide applicability in improving the performance of modern MaxSAT solvers.

The dramatic improvements in Boolean satisfiability (SAT) solving since the turn of the millennium have made it possible to leverage state-of-the-art conflict-driven clause learning (CDCL) solvers for many combinatorial problems in academia and industry, and the use of proof logging has played a crucial role in increasing the confidence that the results these solvers produce are correct. However, the fact that SAT proof logging is performed in conjunctive normal form (CNF) clausal format means that it has not been possible to extend guarantees of correctness to the use of SAT solvers for more expressive combinatorial paradigms, where the first step is an unverified translation of the input to CNF. In this work, we show how cutting-planes-based reasoning can provide proof logging for solvers that translate pseudo-Boolean (a.k.a. 0-1 integer linear) decision problems to CNF and then run CDCL. To support a wide range of encodings, we provide a uniform and easily extensible framework for proof logging of CNF translations. We are hopeful that this is just a first step towards providing a unified proof logging approach that will also extend to maximum satisfiability (MaxSAT) solving and pseudo-Boolean optimization in general.

Incomplete algorithms for the Maximum Satisfiability (MaxSAT) problem use a hill climbing approach in tandem with various mechanisms that prevent search stagnation. These solvers’ conflicting goals of maintaining search mobility while discovering high quality solutions constitute an exploration-exploitation dilemma, a problem which has been tackled with great success in recent years using Monte-Carlo Tree Search (MCTS) methods. We apply MCTS to the domain of MaxSAT using various stochastic local search (SLS) algorithms for leaf node value estimation, thus offering a novel hybrid alternative to established complete and incomplete solution techniques. Our algorithm outperforms baseline SLS algorithms like Walksat and Novelty on most problem instances from the 2015 MaxSAT Evaluation. It also outdoes CCLS, a state-of-the-art incomplete MaxSAT solver, on a number of challenging industrial instances from the 2015 MaxSAT Evaluation.

Boolean satisfiability (SAT) solvers are at the core of efficient approaches for solving a vast multitude of practical problems. Moreover, albeit targeting an NP-complete problem, SAT solvers are increasingly used for tackling problems beyond NP. Despite the success of SAT in practice, modeling with SAT and more importantly implementing SAT-based problem solving solutions is often a difficult and error-prone task. This paper proposes the PySAT toolkit, which enables fast Python-based prototyping using SAT oracles and SAT-related technology. PySAT provides a simple API for working with a few state-of-the-art SAT oracles and also integrates a number of cardinality constraint encodings, all aiming at simplifying the prototyping process. Experimental results presented in the paper show that PySAT-based implementations can be as efficient as those written in a low-level language.

In real-time systems, where tasks have timing requirements, once the workload exceeds the system’s capacity, missed due dates may cause system overload. In this situation, finding an optimal scheduling that minimizes the cumulative values of late tasks is critical in both theory and practice. Recently, formalizing scheduling problems as a class of generalized problems, such as Satisfiability Modulo Theory (SMT) and Maximum Satisfiability (MaxSAT), has been receiving immense concern. Enlightened by the high efficiency of these satisfiability-based methods, this paper formulates the single-machine scheduling problem of minimizing the total weight of late tasks as a Weighted Partial Maximum (WPM) Satisfiability problem. In the formulation, scheduling features are encoded as rigidly enforced hard clauses and the scheduling objective is treated as a set of weighted soft ones. Then an off-the-shelf WPM solver is exploited to maximize the total weight of the satisfied soft clauses, provided that all the hard clauses are satisfied. Experimental results demonstrate that, compared with the existing satisfiability-based methods, the proposed method significantly improves the efficiency of identifying the optimal schedule. Moreover, we make minor changes to apply the WPM formulation to parallel-machine scheduling, showing that the proposed method is sufficiently flexible and well scalable.

Modern satisfiability (SAT) solvers provide an efficient implementation of classical propositional logic. Their input language, however, is based on the conjunctive normal form (CNF) of propositional formulas. To use SAT solver technology in practice, a user must create the input clauses in one way or another. A typical approach is to write a procedural program that generates formulas on the basis of some input data relevant for the problem domain and translates them into CNF. In this paper, we propose a declarative approach where the intended clauses are specified in terms of rules in analogy to answer set programming (ASP). This allows the user to write first-order specifications for intended clauses in a schematic way by exploiting term variables. We develop a formal framework required to define the semantics of such specifications. Moreover, we provide an implementation harnessing state-of-the-art ASP grounders to accomplish the grounding step of clauses. As a result, we obtain a general-purpose clause-level grounding approach for SAT solvers. Finally, we illustrate the capabilities of our specification methodology in terms of combinatorial and application problems.

Query-answering (QA) problems have attracted wider interest recently, owing partly to emerging applications involving integration between rules and ontologies in the Semantic Web's layered architecture. Success of using satisfiability (SAT) solvers as low-level solvers for dealing with several kinds of problems motivates us to apply SAT solvers to a class of QA problems. To find all elements of an answer set, a variant of a usual SAT solver, called an all-solution satisfiability (All-SAT) solver, is used for determining all models of a given input set of propositional clauses. Using an All-SAT solver, we propose a procedure for solving QA problems. The procedure generates input for an All-SAT solver from a high-level description of a given QA problem by problem transformation in a clause space that includes set-bounded variables. Based on the equivalent transformation principle, the correctness of the proposed procedure is shown.

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.