Symmetries are intrinsic to many combinatorial problems including Boolean Satisfiability (SAT) and Constraint Programming (CP). In SAT, the identification of symmetry breaking predicates (SBPs) is a well-known, often effective, technique for solving hard problems. The identification of SBPs in SAT has been the subject of significant improvements in recent years, resulting in more compact SBPs and more effective algorithms. The identification of SBPs has also been applied to pseudo-Boolean (PB) constraints, showing that symmetry breaking can also be an effective technique for PB constraints. This paper extends further the application of SBPs, and shows that SBPs can be identified and used in Maximum Satisfiability (MaxSAT), as well as in its most well-known variants, including partial MaxSAT, weighted MaxSAT and weighted partial MaxSAT. As with SAT and PB, symmetry breaking predicates for MaxSAT and variants are shown to be effective for a representative number of problem domains, allowing solving problem instances that current state of the art MaxSAT solvers could not otherwise solve.
Cairo University（开罗大学）；Faculty of Science（理学院）；Department of Mathematics（数学系）
In this paper, we extend the Maximum Satisfiability (MaxSAT) problem to Łukasiewicz logic. The MaxSAT problem for a set of formulae Φ is the problem of finding an assignment to the variables in Φ that satisfies the maximum number of formulae. Three possible solutions (encodings) are proposed to the new problem: (1) Disjunctive Linear Relations (DLRs), (2) Mixed Integer Linear Programming (MILP) and (3) Weighted Constraint Satisfaction Problem (WCSP). Like its Boolean counterpart, the extended fuzzy MaxSAT will have numerous applications in optimization problems that involve vagueness.
Maximum Satisfiability (MaxSAT) is a well-known optimization pro- blem, with several practical applications. The most widely known MAXS AT algorithms are ineffective at solving hard problems instances from practical application domains. Recent work proposed using efficient Boolean Satisfiability (SAT) solvers for solving the MaxSAT problem, based on identifying and eliminating unsatisfiable subformulas. However, these algorithms do not scale in practice. This paper analyzes existing MaxSAT algorithms based on unsatisfiable subformula identification. Moreover, the paper proposes a number of key…
DSO National Laboratories（DSO国家实验室）；National University of Singapore（新加坡国立大学）
The survey propagation (SP) algorithm has been shown to work well on large instances of the random 3-SAT problem near its phase transition. It was shown that SP estimates marginals over covers that represent clusters of solutions. The SP-y algorithm generalizes SP to work on the maximum satisfiability (Max-SAT) problem, but the cover interpretation of SP does not generalize to SP-y. In this paper, we formulate the relaxed survey propagation (RSP) algorithm, which extends the SP algorithm to apply to the weighted Max-SAT problem. We show that RSP has an interpretation of estimating marginals ov…
United Technologies Research Center（联合技术研究中心）；SRI International（SRI国际）
Given a Boolean formula $ϕ(x)$ in conjunctive normal form (CNF), the density of states counts the number of variable assignments that violate exactly $e$ clauses, for all values of $e$. Thus, the density of states is a histogram of the number of unsatisfied clauses over all possible assignments. This computation generalizes both maximum-satisfiability (MAX-SAT) and model counting problems and not only provides insight into the entire solution space, but also yields a measure for the \emph{hardness} of the problem instance. Consequently, in real-world scenarios, this problem is typically infeas…
University of Helsinki（赫尔辛基大学）；Aalto University（阿尔托大学）；Helsinki Institute of Information Technology（赫尔辛基信息技术研究所）；Griffith University（格里菲斯大学）；Åbo Akademi University（奥布学术大学）
We investigate the problem of learning the structure of a Markov network from data. It is shown that the structure of such networks can be described in terms of constraints which enables the use of existing solver technology with optimization capabilities to compute optimal networks starting from initial scores computed from the data. To achieve efficient encodings, we develop a novel characterization of Markov network structure using a balancing condition on the separators between cliques forming the network. The resulting translations into propositional satisfiability and its extensions such…
Boolean Satisfiability solvers have gone through dramatic improvements in their performances and scalability over the last few years by considering symmetries. It has been shown that by using graph symmetries and generating symmetry breaking predicates (SBPs) it is possible to break symmetries in Conjunctive Normal Form (CNF). The SBPs cut down the search space to the nonsymmetric regions of the space without affecting the satisfiability of the CNF formula. The symmetry breaking predicates are created by representing the formula as a graph, finding the graph symmetries and using some symmetry …
University of Oxford（牛津大学）；INESC-ID；Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院）
Maximum Satisfiability (MaxSAT) is an optimization variant of the Boolean Satisfiability (SAT) problem. In general, MaxSAT algorithms perform a succession of SAT solver calls to reach an optimum solution making extensive use of cardinality constraints. Many of these algorithms are non-incremental in nature, i.e. at each iteration the formula is rebuilt and no knowledge is reused from one iteration to another. In this paper, we exploit the knowledge acquired across iterations using novel schemes to use cardinality constraints in an incremental fashion. We integrate these schemes with several Ma…
It was shown before that the NP-hard problem of deterministic finite automata (DFA) identification can be effectively translated to Boolean satisfiability (SAT). Modern SAT-solvers can tackle hard DFA identification instances efficiently. We present a technique to reduce the problem search space by enforcing an enumeration of DFA states in depth-first search (DFS) or breadth-first search (BFS) order. We propose symmetry breaking predicates, which can be added to Boolean formulae representing various DFA identification problems. We show how to apply this technique to DFA identification from bot…
INESC-ID；Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院）
The most successful parallel SAT and MaxSAT solvers follow a portfolio approach, where each thread applies a different algorithm (or the same algorithm configured differently) to solve a given problem instance. The main goal of building a portfolio is to diversify the search process being carried out by each thread. As soon as one thread finishes, the instance can be deemed solved. In this paper we present a new open source distributed solver for MaxSAT solving that addresses two issues commonly found in multicore parallel solvers, namely memory contention and scalability. Preliminary results …
Laboratoire de Théorie des Communications（通信理论实验室）；Faculté Informatique et Communications（计算机与通信学院）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
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 deep 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 anal…
Carnegie Mellon University（卡内基梅隆大学）；IIT Hyderabad（印度理工学院海得拉巴分校）；INESC-ID；Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院）
To celebrate the first 25 years of the International Conference on Principles and Practice of Constraint Programming (CP) the editors invited the authors of the most cited paper of each year to write a commentary on their paper. This report describes our reflections on the CP 2014 paper "Incremental Cardinality Constraints for MaxSAT" and its impact on the Maximum Satisfiability community and beyond.
Huazhong University of Science and Technology（华中科技大学）；Université de Picardie Jules Verne（皮卡第儒勒·凡尔纳大学）
Although Path-Relinking is an effective local search method for many combinatorial optimization problems, its application is not straightforward in solving the MAX-SAT, an optimization variant of the satisfiability problem (SAT) that has many real-world applications and has gained more and more attention in academy and industry. Indeed, it was not used in any recent competitive MAX-SAT algorithms in our knowledge. In this paper, we propose a new local search algorithm called IPBMR for the MAX-SAT, that remedies the drawbacks of the Path-Relinking method by using a careful combination of three …
Machine learning (ML) is ubiquitous in modern life. Since it is being deployed in technologies that affect our privacy and safety, it is often crucial to understand the reasoning behind its decisions, warranting the need for explainable AI. Rule-based models, such as decision trees, decision lists, and decision sets, are conventionally deemed to be the most interpretable. Recent work uses propositional satisfiability (SAT) solving (and its optimization variants) to generate minimum-size decision sets. Motivated by limited practical scalability of these earlier methods, this paper proposes a no…
Reliable Reasoning with Large Language Models via Preference-Based Maximum Satisfiability
Artificial Intelligence Research Institute (IIIA) Consejo Superior de Investigaciones Científicas (CSIC)（人工智能研究所（IIIA）西班牙国家科学研究委员会（CSIC））；Department of Computer Science University of Oxford（计算机科学系牛津大学）；Institut de Robòtica i Informàtica Industrial (IRI-CSIC-UPC)（机器人与信息工业研究所（IRI-CSIC-UPC））
University of Michigan（密歇根大学）；American University in Sharjah（沙迦美国大学）
Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean …
School of Software, Dalian University of Technology（大连理工大学软件学院）；The State Key Laboratory of Computer Science, Institute of Software, CAS（中国科学院软件研究所计算机科学国家重点实验室）
The weighted Maximum Satisfiability problem (weighted MAX-SAT) is a NP-hard problem with numerous applications arising in artificial intelligence. As an efficient tool for heuristic design, the backbone has been applied to heuristics design for many NP-hard problems. In this paper, we investigated the computational complexity for retrieving the backbone in weighted MAX-SAT and developed a new algorithm for solving this problem. We showed that it is intractable to retrieve the full backbone under the assumption that . Moreover, it is intractable to retrieve a fixed fraction of the backbone as w…
The Maximum Satisfiability (MaxSAT) problem is the problem of finding a truth assignment that maximizes the number of satisfied clauses of a given Boolean formula in Conjunctive Normal Form (CNF). Many exact solvers for MaxSAT have been developed during recent years, and many of them were presented in the well-known SAT conference. Algorithms for MaxSAT generally fall into two categories: (1) branch and bound algorithms and (2) algorithms that use successive calls to a SAT solver (SAT- based), which this paper in on. In practical problems, SAT-based algorithms have been shown to be more effici…
Université Catholique de Louvain（鲁汶天主教大学）；University of Toronto Scarborough（多伦多大学士嘉堡分校）；Rotman School of Management（罗特曼管理学院）
This paper presents and evaluates two pruning techniques to reinforce the efficiency of constraint optimization solvers based on multi-valued decision-diagrams (MDD). It adopts the branch-and-bound framework proposed by Bergman et al. in 2016 to solve dynamic programs to optimality. In particular, our paper presents and evaluates the effectiveness of the local-bound (LocB) and rough upper-bound pruning (RUB). LocB is a new and effective rule that leverages the approximate MDD structure to avoid the exploration of non-interesting nodes. RUB is a rule to reduce the search space during the develo…
KU Leuven（鲁汶大学）；Helsinki Institute for Information Technology HIIT, Aalto University（赫尔辛基信息技术研究所，阿尔托大学）
Symmetry breaking has been proven to be an efficient preprocessing technique for satisfiability solving (SAT). In this paper, we port the state-of-the-art SAT symmetry breaker BreakID to answer set programming (ASP). The result is a lightweight tool that can be plugged in between the grounding and the solving phases that are common when modelling in ASP. We compare our tool with sbass, the current state-of-the-art symmetry breaker for ASP.
Enhancing Local Search for MaxSAT with Deep Differentiation Clause Weighting
School of Computer Science, South China Normal University, Guangzhou, China（计算机科学学院，华南师范大学，广州，中国）；School of Artificial Intelligence, South China Normal University, Foshan, China（人工智能学院，华南师范大学，佛山，中国）
Carnegie Mellon University（卡内基梅隆大学）；University of Southern California（南加州大学）；Bosch Center for Artificial Intelligence（博世人工智能中心）
Integrating logical reasoning within deep learning architectures has been a major goal of modern AI systems. In this paper, we propose a new direction toward this goal by introducing a differentiable (smoothed) maximum satisfiability (MAXSAT) solver that can be integrated into the loop of larger deep learning systems. Our (approximate) solver is based upon a fast coordinate descent approach to solving the semidefinite program (SDP) associated with the MAXSAT problem. We show how to analytically differentiate through the solution to this SDP and efficiently solve the associated backward pass. W…
University of Lleida（莱里达大学）；Artificial Intelligence Research Institute (IIIA, CSIC)（人工智能研究所（西班牙国家研究委员会IIIa研究所））
We present a Satisfiability (SAT)-based approach for building Mixed Covering Arrays with Constraints of minimum length, referred to as the Covering Array Number problem. This problem is central in Combinatorial Testing for the detection of system failures. In particular, we show how to apply Maximum Satisfiability (MaxSAT) technology by describing efficient encodings for different classes of complete and incomplete MaxSAT solvers to compute optimal and suboptimal solutions, respectively. Similarly, we show how to solve through MaxSAT technology a closely related problem, the Tuple Number probl…
State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所计算机科学国家重点实验室）；Laboratory of Parallel Software and Computational Science, Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所并行软件与计算科学实验室）；University of Chinese Academy of Sciences（中国科学院大学）
With the rapid development of deep learning techniques, various recent work has tried to apply graph neural networks (GNNs) to solve NP-hard problems such as Boolean Satisfiability (SAT), which shows the potential in bridging the gap between machine learning and symbolic reasoning. However, the quality of solutions predicted by GNNs has not been well investigated in the literature. In this paper, we study the capability of GNNs in learning to solve Maximum Satisfiability (MaxSAT) problem, both from theoretical and practical perspectives. We build two kinds of GNN models to learn the solution o…
Due to the good performance of current SAT (satisfiability) and Max-SAT (maximum ssatisfiability) solvers, many real-life optimization problems such as scheduling can be solved by encoding them into Max-SAT. In this paper we tackle the course timetabling problem of the department of mathematics, Cairo University by encoding it into Max-SAT. Generating timetables for the department by hand has proven to be cumbersome and the generated timetable almost always contains conflicts. We show how the constraints can be modelled as a Max-SAT instance.
In Bayesian inference, the maximum a posteriori (MAP) problem combines the most probable explanation (MPE) and marginalization (MAR) problems. The counterpart in propositional logic is the exist-random stochastic satisfiability (ER-SSAT) problem, which combines the satisfiability (SAT) and weighted model counting (WMC) problems. Both MAP and ER-SSAT have the form $\operatorname{argmax}_X \sum_Y f(X, Y)$, where $f$ is a real-valued function over disjoint sets $X$ and $Y$ of variables. These two optimization problems request a value assignment for the $X$ variables that maximizes the weighted su…
