MaxSAT Solvers Research Articles

A fast test compaction method using dedicated Pure MaxSAT solver embedded in DFT flow

Minimizing the testing cost is crucial in the context of the design for test (DFT) flow. In our observation, the test patterns generated by ATPG tools in test compression mode still contain redundancy. To tackle this obstacle, we propose a post-flow static test compaction method that utilizes a partial fault dictionary instead of a full fault dictionary to sharply reduce time and memory overhead, and leverages a dedicated Pure MaxSAT solver to re-compact the test patterns generated by ATPG tools. We also observe that ATPG tools offer a more comprehensive selection of candidate patterns for compaction in the “n-detect” mode, leading to superior compaction efficiency. In our experiments conducted on benchmark circuits ISCAS89, ITC99, and an open-source RISC-V CPU, we employed two methodologies. For commercial tool, we utilized a non-intrusive approach, while we adopted an intrusive method for open-source ATPG. Under the non-intrusive approach, our method achieved a maximum reduction of 34.69% in pattern count and a maximum 29.80% decrease in test cycles as evaluated by a leading commercial tool. Meanwhile, under the intrusive approach, our method attained a maximum 71.90% reduction in pattern count as evaluated by an open-source ATPG tool. Notably, fault coverage remained unchanged throughout the experiments. Furthermore, our approach demonstrates improved performance compared with existing methods.

Enhance Diversified Top-k MaxSAT Solving by Incorporating New Strategy for Generating Diversified Initial Assignments (Student Abstract)

The Diversified Top-k MaxSAT (DTKMS) problem is an extension of MaxSAT. The objective of DTKMS is to find k feasible assignments of a given formula, such that each assignment satisfies all hard clauses and the k assignments together satisfy the maximum number of soft clauses. This paper presents a local search algorithm, DTKMS-DIA, which incorporates a new approach to generating initial assignments. Experimental results indicate that DTKMS-DIA can achieve attractive performance on 826 instances compared with state-of-the-art solvers.

Theoretical and Empirical Analysis of Cost-Function Merging for Implicit Hitting Set WCSP Solving

The Implicit Hitting Set (HS) approach has shown very effective for MaxSAT solving. However, only preliminary promising results have been obtained for the very similar Weighted CSP framework. In this paper we contribute towards both a better theoretical understanding of the HS approach and a more effective HS-based solvers for WCSP. First, we bound the minimum number of iterations of HS thanks to what we call distinguished cores. Then, we show a source of inefficiency by introducing two simple problems where HS is unfeasible. Next, we propose two reformulation methods that merge cost-functions to overcome the problem. We provide a theoretical analysis that quantifies the magnitude of the improvement of each method with respect to the number of iterations of the algorithm. In particular, we show that the reformulations can bring an exponential number of iterations down to a constant number in our working examples. Finally, we complement our theoretical analysis with two sets of experiments. First, we show that our results are aligned with real executions. Second, and most importantly, we conduct experiments on typical benchmark problems and show that cost-function merging may be heuristically applied and it may accelerate HS algorithms by several orders of magnitude. In some cases, it even outperforms state-of-the-art solvers.

Farsighted Probabilistic Sampling: A General Strategy for Boosting Local Search MaxSAT Solvers

Local search has been demonstrated as an efficient approach for two practical generalizations of the MaxSAT problem, namely Partial MaxSAT (PMS) and Weighted PMS (WPMS). In this work, we observe that most local search (W)PMS solvers usually flip a single variable per iteration. Such a mechanism may lead to relatively low-quality local optimal solutions, and may limit the diversity of search directions to escape from local optima. To address this issue, we propose a general strategy, called farsighted probabilistic sampling (FPS), to replace the single flipping mechanism so as to boost the local search (W)PMS algorithms. FPS considers the benefit of continuously flipping a pair of variables in order to find higher-quality local optimal solutions. Moreover, FPS proposes an effective approach to escape from local optima by preferring the best to flip among the best sampled single variable and the best sampled variable pair. Extensive experiments demonstrate that our proposed FPS strategy significantly improves the state-of-the-art (W)PMS solvers, and FPS has an excellent generalization capability to various local search MaxSAT solvers.

Conformance checking artefacts through weighted partial MaxSAT

Conformance checking techniques are based on the computation of artefacts which enable the reasoning about observed behaviour of processes and their models. A family of these related artefacts, alignments, multi- and anti-alignments are defined over a distance function between observed behaviour and model replay, both represented as words from a given alphabet. Encoding this distance function, commonly the Levenshtein distance, into MaxSAT enables us to take advantage of state-of-the-art MaxSAT solvers. In this paper we provide an improvement upon previous SAT encodings for the alignment artefact family. We achieve a reduction in the resulting formula size while also obtaining good performance results on the computation of the artefacts. Additionally, we provide its direct extension that allows to easily change the cost function for the alignment computation and show the implementation of two particular examples from the literature: prefix matching prioritization and synchronization maximization.

Proofs and Certificates for Max-SAT

Current Max-SAT solvers are able to efficiently compute the optimal value of an input instance but they do not provide any certificate of its validity. In this paper, we present a tool, called MS-Builder, which generates certificates for the Max-SAT problem in the particular form of a sequence of equivalence-preserving transformations. To generate a certificate, MS-Builder iteratively calls a SAT oracle to get a SAT resolution refutation which is handled and adapted into a sound refutation for Max-SAT. In particular, we prove that the size of the computed Max-SAT refutation is linear with respect to the size of the initial refutation if it is semi-read-once, tree-like regular, tree-like or semi-tree-like. Additionally, we propose an extendable tool, called MS-Checker, able to verify the validity of any Max-SAT certificate using Max-SAT inference rules. Both tools are evaluated on the unweighted and weighted benchmark instances of the 2020 Max-SAT Evaluation.

Clausal Forms in MaxSAT and MinSAT

We tackle the problem of reducing non-clausal MaxSAT and MinSAT to clausal MaxSAT and MinSAT. Our motivation is twofold: (i) the clausal form transformations used in SAT are unsound for MaxSAT and MinSAT, because they do not preserve the minimum or maximum number of unsatisfied clauses, and (ii) the state-of-the-art MaxSAT and MinSAT solvers require as input a multiset of clauses. The main contribution of this paper is the definition of three different cost-preserving transformations. Two transformations extend the usual equivalence preserving transformation used in SAT to MaxSAT and MinSAT. The third one extends the well-known Tseitin transformation. Furthermore, we report on an empirical comparison of the performance of the proposed transformations when solved with a state-of-the-art MaxSAT solver.

Synthesis of Cost-Optimal Multi-Agent Systems for Resource Allocation

Multi-agent systems for resource allocation (MRAs) have been introduced as a concept for modelling competitive resource allocation problems in distributed computing. An MRA is composed of a set of agents and a set of resources. Each agent has goals in terms of allocating certain resources. For MRAs it is typically of importance that they are designed in a way such that there exists a strategy that guarantees that all agents will achieve their goals. The corresponding model checking problem is to determine whether such a winning strategy exists or not, and the synthesis problem is to actually build the strategy. While winning strategies ensure that all goals will be achieved, following such strategies does not necessarily involve an optimal use of resources. In this paper, we present a technique that allows to synthesise cost-optimal solutions to distributed resource allocation problems. We consider a scenario where system components such as agents and resources involve costs. A multi-agent system shall be designed that is cost-minimal but still capable of accomplishing a given set of goals. Our approach synthesises a winning strategy that minimises the cumulative costs of the components that are required for achieving the goals. The technique is based on a propositional logic encoding and a reduction of the synthesis problem to the maximum satisfiability problem (Max-SAT). Hence, a Max-SAT solver can be used to perform the synthesis. From a truth assignment that maximises the number of satisfied clauses of the encoding a cost-optimal winning strategy as well as a cost-optimal system can be immediately derived.

Negative Learning Ant Colony Optimization for MaxSAT

Recently, a new negative learning variant of ant colony optimization (ACO) has been used to successfully tackle a range of combinatorial optimization problems. For providing stronger evidence of the general applicability of negative learning ACO, we investigate how it can be adapted to solve the Maximum Satisfiability problem (MaxSAT). The structure of MaxSAT is different from the problems considered to date and there exists only a few ACO approaches for MaxSAT. In this paper, we describe three negative learning ACO variants. They differ in the way in which sub-instances are solved at each algorithm iteration to provide negative feedback to the main ACO algorithm. In addition to using IBM ILOG CPLEX, two of these variants use existing MaxSAT solvers for this purpose. The experimental results show that the proposed negative learning ACO variants significantly outperform the baseline ACO as well as IBM ILOG CPLEX and the two MaxSAT solvers. This result is of special interest because it shows that negative learning ACO can be used to improve over the results of existing solvers by internally using them to solve smaller sub-instances.

The Impact of Implied Constraints on MaxSAT B2B Instances

The B2B scheduling optimization problem consists of finding a schedule of a set of meetings between pairs of participants, minimizing their number of idle time periods. Recent works have shown that SAT-based approaches are state-of-the-art on this problem. One interesting feature of such approaches is the use of implied constraints. In this work, we provide an experimental setting to study the impact of using these implied constraints in MaxSAT B2B instances. To this purpose and due to the reduced number of existing real-world B2B instances, we propose a random B2B instance generation model, which reproduces certain features of these problems. In our experimental analysis, we show that the impact of using some implied constraints in the MaxSAT encodings depends on the characteristics of the problem, and we also analyze the benefits of combining them. Finally, we give some insights on how a MaxSAT solver is able to exploit these implied constraints.

Boosting branch-and-bound MaxSAT solvers with clause learning

The Maximum Satisfiability Problem, or MaxSAT, offers a suitable problem solving formalism for combinatorial optimization problems. Nevertheless, MaxSAT solvers implementing the Branch-and-Bound (BnB) scheme have not succeeded in solving challenging real-world optimization problems. It is widely believed that BnB MaxSAT solvers are only superior on random and some specific crafted instances. At the same time, SAT-based MaxSAT solvers perform particularly well on real-world instances. To overcome this shortcoming of BnB MaxSAT solvers, this paper proposes a new BnB MaxSAT solver called MaxCDCL. The main feature of MaxCDCL is the combination of clause learning of soft conflicts and an efficient bounding procedure. Moreover, the paper reports on an experimental investigation showing that MaxCDCL is competitive when compared with the best performing solvers of the 2020 MaxSAT Evaluation. MaxCDCL performs very well on real-world instances, and solves a number of instances that other solvers cannot solve. Furthermore, MaxCDCL, when combined with the best performing MaxSAT solvers, solves the highest number of instances of a collection from all the MaxSAT evaluations held so far.

Using MaxSAT for Efficient Explanations of Tree Ensembles

Tree ensembles (TEs) denote a prevalent machine learning model that do not offer guarantees of interpretability, that represent a challenge from the perspective of explainable artificial intelligence. Besides model agnostic approaches, recent work proposed to explain TEs with formally-defined explanations, which are computed with oracles for propositional satisfiability (SAT) and satisfiability modulo theories. The computation of explanations for TEs involves linear constraints to express the prediction. In practice, this deteriorates scalability of the underlying reasoners. Motivated by the inherent propositional nature of TEs, this paper proposes to circumvent the need for linear constraints and instead employ an optimization engine for pure propositional logic to efficiently handle the prediction. Concretely, the paper proposes to use a MaxSAT solver and exploit the objective function to determine a winning class. This is achieved by devising a propositional encoding for computing explanations of TEs. Furthermore, the paper proposes additional heuristics to improve the underlying MaxSAT solving procedure. Experimental results obtained on a wide range of publicly available datasets demonstrate that the proposed MaxSAT-based approach is either on par or outperforms the existing reasoning-based explainers, thus representing a robust and efficient alternative for computing formal explanations for TEs.

Proof Complexity for the Maximum Satisfiability Problem and its Use in SAT Refutations

Abstract MaxSAT, the optimization version of the well-known SAT problem, has attracted a lot of research interest in the past decade. Motivated by the many important applications and inspired by the success of modern SAT solvers, researchers have developed many MaxSAT solvers. Since most research is algorithmic, its significance is mostly evaluated empirically. In this paper, we want to address MaxSAT from the more formal point of view of proof complexity. With that aim, we start providing basic definitions and proving some basic results. Then we analyse the effect of adding split and virtual, two original inference rules, to MaxSAT resolution. We show that each addition makes the resulting proof system stronger, even when virtual is restricted to empty clauses ($0$-virtual). We also analyse the power of our proof systems in the particular case of SAT refutations. We show that our strongest system, ResSV, is equivalent to circular and dual rail with split. We also analyse empirically some known gadget-based reformulations. Our results seem to indicate that the advantage of these three seemingly different systems over general resolution comes mainly from their ability of augmenting the original formula with hypothetical inconsistencies, as captured in a very simple way by the virtual rule.

MAXSAT Heuristics for Cost Optimal Planning

The cost of an optimal delete relaxed plan, known as h+, is a powerful admissible heuristic but is in general intractable to compute. In this paper we examine the problem of computing h+ by encoding it as a MAXSAT problem. We develop a new encoding that utilizes constraint generation to support the computation of a sequence of increasing lower bounds on h+. We show a close connection between the computations performed by a recent approach for solving MAXSAT and a hitting set approach recently proposed for computing h+. Using this connection we observe that our MAXSAT computation can be initialized with a set of landmarks computed by LM-cut. By judicious use of MAXSAT solving along with a technique of lazy heuristic evaluation we obtain speedups for finding optimal plans over LM-cut on a number of domains. Our approach enables the exploitation of continued progress in MAXSAT solving, and also makes it possible to consider computing or approximating heuristics that are even more informed that h+ by, for example, adding some information about deletes back into the encoding.

Handling All Unit Propagation Reasons in Exact Max-SAT Solvers

Unit propagation (UP) based method are widely used in Branch and Bound (BnB) Max-SAT solvers for detecting disjoint inconsistent subsets (IS) during the lower bound (LB) estimation. UP consists in assigning to true (propagating) all the literals which appear in unit clauses. The existing implementations of UP only consider the first unit clause causing the assignment of each variable, thus the propagations must be done and undone chronologically to ensure that all the unit clauses are properly exploited. Max-SAT BnB solvers transforms the formulas to ensure IS disjointness. These transformations remove clauses from the formula thus propagations are frequently undone. Since the propagations are undone in chronological order, many useless unassignments and reassignments are performed. We propose in this paper a new unit propagation scheme which considers all the unit clauses causing the assignment of the variables by UP. This new scheme allows to undo propagations in a non-chronological way and thus it reduces the number of redundant propagation steps made by BnB solvers. We also show how the information available with this new scheme can be used to influence the characteristics of the IS built by BnB solvers. We propose a heuristic which aims at reducing their size, and thus improving the quality of the LB estimation. We have implemented the new propagation scheme as well as the IS building heuristic in our solver MSsolver. We present and discuss the results of the experimental study we have performed.

A New Boolean Encoding for MAPF and its Performance with ASP and MaxSAT Solvers

Multi-agent pathfinding (MAPF) is an NP-hard problem. As such, dense maps may be very hard to solve optimally. In such scenarios, compilation-based approaches, via Boolean satisfiability (SAT) and answer set programming (ASP), have proven to be most effective. In this paper, we propose a new encoding for MAPF, which we implement and solve using both ASP and MaxSAT solvers. Our encoding builds on a recent ASP encoding for MAPF but changes the way agent moves are encoded. This allows to represent swap and follow conflicts with binary clauses, which are known to work well along with conflict-based clause learning. For MaxSAT, we study different ways in which we may combine the MSU3 and LSU algorithms for maximum performance. Our results, over grid and warehouse maps, show that the ASP solver scales better when the number of agents is increased on grids with few obstacles, while the MaxSAT solver performs better in scenarios with more obstacles and fewer agents.

Polarity and Variable Selection Heuristics for SAT-Based Anytime MaxSAT

This paper is a system description of the anytime MaxSAT solver TT-Open-WBO-Inc, which won both of the weighted incomplete tracks of MaxSAT Evaluation 2019. We implemented the recently introduced polarity and variable selection heuristics, TORC and TSB, respectively, in the Open-WBO-Inc-BMO algorithm within the open-source anytime MaxSAT solver Open-WBO-Inc. As a result, the solver is substantially more efficient.

Recognizing Multi-Agent Plans When Action Models and Team Plans Are Both Incomplete

Multi-Agent Plan Recognition (MAPR) aims to recognize team structures (which are composed of team plans) from the observed team traces (action sequences) of a set of intelligent agents. In this article, we introduce the problem formulation of MAPR based on partially observed team traces, and present a weighted MAX-SAT–based framework to recognize multi-agent plans from partially observed team traces with the help of two types of auxiliary knowledge to help recognize multi-agent plans, i.e., a library ofincompleteteam plans and a set ofincompleteaction models. Our framework functions with two phases. We first build a set ofhardconstraints that encode the correctness property of the team plans, and a set ofsoftconstraints that encode the optimal utility property of team plans based on the input team trace, incomplete team plans, and incomplete action models. After that, we solve all of the constraints using a weighted MAX-SAT solver and convert the solution to a set of team plans that bestexplainthe structure of the observed team trace. We empirically exhibit both effectiveness and efficiency of our framework in benchmark domains from International Planning Competition (IPC).

A continuous-time MaxSAT solver with high analog performance

Many real-life optimization problems can be formulated in Boolean logic as MaxSAT, a class of problems where the task is finding Boolean assignments to variables satisfying the maximum number of logical constraints. Since MaxSAT is NP-hard, no algorithm is known to efficiently solve these problems. Here we present a continuous-time analog solver for MaxSAT and show that the scaling of the escape rate, an invariant of the solver’s dynamics, can predict the maximum number of satisfiable constraints, often well before finding the optimal assignment. Simulating the solver, we illustrate its performance on MaxSAT competition problems, then apply it to two-color Ramsey number R(m, m) problems. Although it finds colorings without monochromatic 5-cliques of complete graphs on N ≤ 42 vertices, the best coloring for N = 43 has two monochromatic 5-cliques, supporting the conjecture that R(5, 5) = 43. This approach shows the potential of continuous-time analog dynamical systems as algorithms for discrete optimization.

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.