Partial Maximum Satisfiability (PMS) and Weighted Partial Maximum Satisfiability (WPMS) generalize Maximum Satisfiability (MaxSAT), with broad real-world applications. Recent advances in Stochastic Local Search (SLS) algorithms for solving (W)PMS have mainly focused on designing clause weighting schemes. However, existing methods often fail to adequately distinguish between PMS and WPMS, typically employing uniform update strategies for clause weights and overlooking critical structural differences between the two problem types. In this work, we present a novel clause weighting scheme that, for the first time, updates the clause weights of PMS and WPMS instances according to distinct conditions. This scheme also introduces a new initialization method, which better accommodates the unique characteristics of both instance types. Furthermore, we propose a decimation method that prioritizes satisfying unit and hard clauses, effectively complementing our proposed clause weighting scheme. Building on these methods, we develop a new SLS solver for (W)PMS named DeepDist. Experimental results on benchmarks from the anytime tracks of recent MaxSAT Evaluations show that DeepDist outperforms state-of-the-art SLS solvers. Notably, a hybrid solver combining DeepDist with TT-Open-WBO-Inc surpasses the performance of the MaxSAT Evaluation 2024 winners, SPB-MaxSAT-c-Band and SPB-MaxSAT-c-FPS, highlighting the effectiveness of our approach. The code is available at https://github.com/jmhmaxsat/DeepDist

Building on the progress in Boolean satisfiability (SAT) solving over the last decades, maximum satisfiability (MaxSAT) has become a viable approach for solving NP-hard optimization problems, but ensuring correctness of MaxSAT solvers has remained an important concern. For SAT, this is largely a solved problem thanks to the use of proof logging, meaning that solvers emit machine-verifiable proofs of (un)satisfiability to certify correctness. However, for MaxSAT, proof logging solvers have started being developed only very recently. Moreover, these nascent efforts have only targeted the core solving process, ignoring the preprocessing phase where input problem instances can be substantially reformulated before being passed on to the solver proper. In this work, we demonstrate how pseudo-Boolean proof logging can be used to certify the correctness of a wide range of modern MaxSAT preprocessing techniques. By combining and extending the VeriPB and CakePB tools, we provide formally verified, end-to-end proof checking that the input and preprocessed output MaxSAT problem instances have the same optimal value. An extensive evaluation on applied MaxSAT benchmarks shows that our approach is feasible in practice.

MaxSAT is an optimization version of the famous NP-complete Satisfiability problem (SAT). Algorithms for MaxSAT mainly include complete solvers and local search incomplete solvers. In many complete solvers, once a better solution is found, a Soft conflict Pseudo Boolean (SPB) constraint will be generated to enforce the algorithm to find better solutions. In many local search algorithms, clause weighting is a key technique for effectively guiding the search directions. In this paper, we propose to transfer the SPB constraint into the clause weighting system of the local search method, leading the algorithm to better solutions. We further propose an adaptive clause weighting strategy that breaks the tradition of using constant values to adjust clause weights. Based on the above methods, we propose a new local search algorithm called SPB-MaxSAT that provides new perspectives for clause weighting on MaxSAT local search solvers. Extensive experiments demonstrate the excellent performance of the proposed methods.

It has been shown that Maximum Satisfiability (MaxSAT) problem instances can be effectively solved by partitioning the set of soft clauses into several disjoint sets. The partitioning methods can be based on clause weights (e.g., stratification) or based on graph representations of the formula. Afterwards, a merge procedure is applied to guarantee that an optimal solution is found. This paper proposes a new framework called UpMax that decouples the partitioning procedure from the MaxSAT solving algorithms. As a result, new partitioning procedures can be defined independently of the MaxSAT algorithm to be used. Moreover, this decoupling also allows users that build new MaxSAT formulas to propose partition schemes based on knowledge of the problem to be solved. We illustrate this approach using several problems and show that partitioning has a large impact on the performance of unsatisfiability-based MaxSAT algorithms.

As an optimization extension of the famous Boolean Satisfiability (SAT) decision problem, the Maximum Satisfiability (MaxSAT) problem aims at finding a complete assignment of the Boolean variables to satisfy as many clauses as possible in a given propositional formula in Conjunctive Normal Form (CNF) [1]. Partial MaxSAT (PMS) is a variant of MaxSAT where the clauses are divided into hard and soft. PMS aims at maximizing the number of satisfied soft clauses with the constraint that all the hard clauses must be satisfied. Associating a positive weight to each soft clause in PMS results in Weighted PMS (WPMS), whose goal is to maximize the total weight of satisfied soft clauses with the same constraint of PMS that all the hard clauses must be satisfied. Both PMS and WPMS, denoted as (W)PMS, have many practical applications such as planning [2], combinatorial testing [3], group testing [4], timetabling [5], etc. Existing solvers for (W)PMS can be divided into complete and incomplete according to whether their solutions have optimality guarantees.

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.

We address Partial MaxSAT (PMS) and Weighted PMS (WPMS), two practical generalizations of the MaxSAT problem, and propose a local search algorithm called BandMaxSAT, that applies a multi-armed bandit to guide the search direction, for these problems. The bandit in our method is associated with all the soft clauses in the input (W)PMS instance. Each arm corresponds to a soft clause. The bandit model can help BandMaxSAT to select a good direction to escape from local optima by selecting a soft clause to be satisfied in the current step, that is, selecting an arm to be pulled. We further propose an initialization method for (W)PMS that prioritizes both unit and binary clauses when producing the initial solutions. Extensive experiments demonstrate that BandMaxSAT significantly outperforms the state-of-the-art (W)PMS local search algorithm SATLike3.0. Specifically, the number of instances in which BandMaxSAT obtains better results is about twice that obtained by SATLike3.0. We further combine BandMaxSAT with the complete solver TT-Open-WBO-Inc. The resulting solver BandMaxSAT-c also outperforms some of the best state-of-the-art complete (W)PMS solvers, including SATLike-c, Loandra and TT-Open-WBO-Inc.

Partial MaxSAT (PMS) and Weighted Partial MaxSAT (WPMS) are both practical generalizations to the typical combinatorial problem of MaxSAT. In this work, we propose an effective farsighted probabilistic sampling based local search algorithm called FPS for solving these two problems, denoted as (W)PMS. The FPS algorithm replaces the mechanism of flipping a single variable per iteration step, that is widely used in existing (W)PMS local search algorithms, with the proposed farsighted local search strategy, and provides higher-quality local optimal solutions. The farsighted strategy employs the probabilistic sampling technique that allows the algorithm to look-ahead widely and efficiently. In this way, FPS can provide more and better search directions and improve the performance without reducing the efficiency. Extensive experiments on all the benchmarks of (W)PMS problems from the incomplete track of recent four years of MaxSAT Evaluations demonstrate that our method significantly outperforms SATLike3.0, the state-of-the-art local search algorithm, for solving both the PMS and WPMS problems. We furthermore do comparison with the extended solver of SATLike, SATLike-c, which is the champion of three categories among the total four (PMS and WPMS categories, each associated with two time limits) of the incomplete track in the recent MaxSAT Evaluation (MSE2021). We replace the local search component in SATLike-c with the proposed farsighted sampling local search approach, and the resulting solver FPS-c also outperforms SATLike-c for solving both the PMS and WPMS problems.

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 problem, which we extend to incorporate constraints. For this problem, we additionally provide a new MaxSAT-based incomplete algorithm. The extensive experimental evaluation we carry out on the available Mixed Covering Arrays with Constraints benchmarks and the comparison with state-of-the-art tools confirm the good performance of our approaches.

In this paper, we present a novel MaxSAT-based technique to compute Maximum Probability Minimal Cut Sets (MPMCSs) in fault trees. We model the MPMCS problem as a Weighted Partial MaxSAT problem and solve it using a parallel SAT-solving architecture. The results obtained with our open source tool indicate that the approach is effective and efficient.