Stochastic Local Search Solvers Research Articles

NuWLS: Improving Local Search for (Weighted) Partial MaxSAT by New Weighting Techniques

Maximum Satisfiability (MaxSAT) is a prototypical constraint optimization problem, and its generalized version is the (Weighted) Partial MaxSAT problem, denoted as (W)PMS, which deals with hard and soft clauses. Considerable progress has been made on stochastic local search (SLS) algorithms for solving (W)PMS, which mainly focus on clause weighting techniques. In this work, we identify two issues of existing clause weighting techniques for (W)PMS, and propose two ideas correspondingly. First, we observe that the initial values of soft clause weights have a big effect on the performance of the SLS solver for solving (W)PMS, and propose a weight initialization method. Second, we propose a new clause weighting scheme that for the first time employs different conditions for updating hard and soft clause weights. Based on these two ideas, we develop a new SLS solver for (W)PMS named NuWLS. Through extensive experiments, NuWLS performs much better than existing SLS solvers on all 6 benchmarks from the incomplete tracks of MaxSAT Evaluations (MSEs) 2019, 2020, and 2021. In terms of the number of winning instances, NuWLS outperforms state-of-the-art SAT-based incomplete solvers on all the 6 benchmarks. More encouragingly, a hybrid solver that combines NuWLS and an SAT-based solver won all four categories in the incomplete track of the MaxSAT Evaluation 2022.

Toward an Understanding of Long-tailed Runtimes of SLS Algorithms

The satisfiability problem (SAT) is one of the most famous problems in computer science. Traditionally, its NP-completeness has been used to argue that SAT is intractable. However, there have been tremendous practical advances in recent years that allow modern SAT solvers to solve instances with millions of variables and clauses. A particularly successful paradigm in this context is stochastic local search (SLS). In most cases, there are different ways of formulating the underlying SAT problem. While it is known that the precise formulation of the problem has a significant impact on the runtime of solvers, finding a helpful formulation is generally non-trivial. The recently introduced GapSAT solver [Lorenz and Wörz 2020] demonstrated a successful way to improve the performance of an SLS solver on average by learning additional information, which logically entails from the original problem. Still, there were also cases in which the performance slightly deteriorated. This justifies in-depth investigations into how learning logical implications affects runtimes for SLS algorithms. In this work, we propose a method for generating logically equivalent problem formulations, generalizing the ideas of GapSAT . This method allows a rigorous mathematical study of the effect on the runtime of SLS SAT solvers. Initially, we conduct empirical investigations. If the modification process is treated as random, then Johnson SB distributions provide a perfect characterization of the hardness. Since the observed Johnson SB distributions approach lognormal distributions, our analysis also suggests that the hardness is long-tailed. As a second contribution, we theoretically prove that restarts are useful for long-tailed distributions. This implies that incorporating additional restarts can further refine all algorithms employing above mentioned modification technique. Since the empirical studies compellingly suggest that the runtime distributions follow Johnson SB distributions, we also investigate this property on a theoretical basis. We succeed in proving that the runtimes for the special case of Schöning’s random walk algorithm [Schöning 2002] are approximately Johnson SB distributed.

Emphasis on the flipping variable: Towards effective local search for hard random satisfiability

Uniform random satisfiability (URS) and hard random satisfiability (HRS) are two important benchmarks for algorithms that solve Boolean satisfiability problems, i.e., SAT solvers, especially for random SAT solvers. Recently, the stochastic local search (SLS) algorithms have made major breakthroughs in URS, resulting in several new state-of-the-art algorithms, e.g., Dimetheus, YalSAT, ProbSAT, and Score2SAT. However, compared to the great progress of SLS on URS, the performance of SLS on HRS lags far behind. In this paper, we propose a new SLS algorithm, named EPEFV for HRS, which employs the extended framework of ProbSAT, and adds a new heuristic method that emphasizes the role of flipping variable, called EFV. EFV focuses on the flipping variables and is based on three components: 1) A new clause weighting scheme focusing on the flipping variable, which is based on a new clause property called UnsatT. By applying this new weighting scheme and a biased random walk, we design a new clause selection mechanism. 2) Design a new scoring function named Uv by combining a novel variable property vUnsatT based on the flipping variable with the commonly used property score.3) A new tie-breaking strategy in the variable selection mechanism based on the new scoring function Uv. Extensive experimental results demonstrate that EPEFV can not only greatly outperforms the state-of-the-art SLS algorithms as well as complete solver competitors on HRS instances, but also can effectively solve URS instances with long clauses. On the contrary, the most advanced SLS solvers, however, can only effectively solve URS instances, while the most advanced complete solvers can only effectively solve HRS instances. At present, no solver can effectively solve both HRS and URS at the same time, which means that the EPEFV can be regarded as the state-of-the-art SLS solver for both HRS instances and URS instances with long clauses. Finally, further empirical analysis confirms the effectiveness of each mechanism underlying the EFV heuristic on HRS instances.

More efficient stochastic local search for satisfiability

Uniform random satisfiability (URS) and hard random satisfiability (HRS) are two significant generalizations of random satisfiability (RS). Recently, great breakthroughs have been made on stochastic local search (SLS) algorithms for uniform RS, resulting in several state-of-the-art algorithms, e.g., Dimetheus, YalSAT, ProbSAT and Score2SAT. However, compared to the great progress of SLS on URS, the performance of SLS on HRS lags far behind. In this paper, we propose two global clause weighting schemes and a new hybrid scoring function called SA based on a linear combination of a property score and property age, and then apply a second-level-biased random walk strategy based on two clause weighting schemes and SA to develop a new SLS solver called BRSAP. To evaluate the performance of BRSAP, we conduct extensive experiments to compare BRSAP with state-of-the-art SLS solvers and complete solvers on HRS instances and URS instances from SAT Competition 2017 and SAT Competition 2018 as well as 4100 generated satisfiable large HRS and URS ones. The experiments illustrate that BRSAP obviously outperforms its competitors, indicating the effectiveness of BRSAP. We also analyze the effectiveness of the underlying ideas in BRSAP.

A Circuit-Level Amoeba-Inspired SAT Solver

AmbSAT (or AmoebaSAT) is a biologically-inspired stochastic local search (SLS) solver to explore solutions to the Boolean satisfiability problem (SAT). AmbSAT updates multiple variables in parallel at every iteration step, and thus AmbSAT can find solutions with a fewer number of iteration steps than some other conventional SLS solvers for a specific set of SAT instances. However, the parallelism of AmbSAT is not compatible with general-purpose microprocessors in that many clock cycles are required to execute each iteration; thus, AmbSAT requires special hardware that can exploit the parallelism of AmbSAT to quickly find solutions. In this paper, we propose a circuit model (hardware-friendly algorithm) that explores solutions to SAT in a similar way to AmbSAT, which we call circuit-level AmbSAT (CL-AmbSAT). We conducted numerical simulation to evaluate the search performance of CL-AmbSAT for a set of randomly generated SAT instances that was designed to estimate the scalability of our approach. Simulation results showed that CL-AmbSAT finds solutions with a fewer iteration number than a powerful SLS solver, ProbSAT, and outperforms even AmbSAT. Since CL-AmbSAT uses simple combinational logic to update variables, CL-AmbSAT can be easily implemented in various hardware.

Focused random walk with probability distribution for SAT with long clauses

Focused random walk (FRW) is one of the most influential paradigm of stochastic local search (SLS) algorithms for the propositional satisfiability (SAT) problem. Recently, an interesting probability distribution (PD) strategy for variable selection was proposed and has been successfully used to improve SLS algorithms, resulting in state-of-the-art solvers. However, most solvers based on the PD strategy only use polynomial function (PoF) to handle the exponential decay and are still unsatisfactory in dealing with medium and huge k-SAT instances at and near the phase transition. The present paper is focused on handling all k-SAT instances with long clauses. Firstly, an extensive empirical study of one state-of-the-art FRW solver WalkSATlm on a wide range of SAT problems is presented with the focus given on fitting the distribution of the break value of variable selected in each step, which turns out to be a Boltzmann function. Using theses case studies as a basis, we propose a pseudo normal function (PNF) to fit the distribution of the break value of variable selected, which is actually a variation of the Boltzmann function. In addition, a new tie-breaking flipping (TBF) strategy is proposed to prevent the same variable from being flipped in consecutive steps. The PNF based PD strategy combined with the TBF strategy lead to a new variable selection heuristic named PNF-TBF. The PNF-TBF heuristic along with a variable allocation value (Vav) function are used to significantly improve ProbSAT, a state-of-the-art SLS solver, leads to a new FRW algorithm dubbed PNFSat, which achieves the state-of-the-art performance on a broad range of huge random 7-SAT instance near the phase transition as demonstrated via the extensive experimental studies. Some further improved versions on top of PNFSat are presented respectively, including PNFSat_alt, which achieves the state-of-the-art performance on the medium 7-SAT instances at the phase transition; PNP as well as an integrated version of these three algorithms, named PDSat, which achieves the state-of-the-art performances on all huge and medium random k-SAT instances with long clauses as demonstrated via the comparative studies using different benchmarks.

Applying aspiration in local search for satisfiability.

The Boolean Satisfiability problem (SAT) is a prototypical NP-complete problem, which has been widely studied due to its significant importance in both theory and applications. Stochastic local search (SLS) algorithms are among the most efficient approximate methods available for solving certain types of SAT instances. The quantitative configuration checking (QCC) heuristic is an effective approach for improving SLS algorithms on solving the SAT problem, resulting in an efficient SLS solver for SAT named Swqcc. In this paper, we focus on combining the QCC heuristic with an aspiration mechanism, and then design a new heuristic called QCCA. On the top of Swqcc, we utilize the QCCA heuristic to develop a new SLS solver dubbed AspiSAT. Through extensive experiments, the results illustrate that, on random 3-SAT instances, the performance of AspiSAT is much better than that of Swqcc and Sparrow, which is an influential and efficient SLS solver for SAT. In addition, we further enhance the original clause weighting schemes employed in Swqcc and AspiSAT, and thus obtain two new SLS solvers called Ptwqcc and AspiPT, respectively. The eperimental results present that both Ptwqcc and AspiPT outperform Swqcc and AspiSAT on random 5-SAT instances, indicating that both QCC and QCCA heuristics are able to cooperate effectively with different clause weighting schemes.

Going deeper with optimal software products selection using many-objective optimization and satisfiability solvers

In search-based software engineering, one actively studied problem is the optimal software product selection from a feature model using multiple (usually more than three) optimization objectives simultaneously. This can be represented as a many-objective optimization problem. The primary goal of solving this problem is to search for diverse and high-quality valid products as rapidly as possible. Previous studies have shown that combining search-based techniques with satisfiability (SAT) solvers was promising for achieving this goal, but it remained open that how different solvers affect the performance of a search algorithm, and that whether the ways to randomize solutions in the solvers make a difference. Moreover, we may need further investigation on the necessity of mixing different types of SAT solving techniques. In this paper, we address the above open research questions by performing a series of empirical studies on 21 features models, most of which are reverse-engineered from industrial software product lines. We examine four conflict-driven clause learning solvers, two stochastic local search solvers, and two different ways to randomize solutions. Experimental results suggest that the performance can be indeed affected by different SAT solvers, and by the ways to randomize solutions in the solvers. This study serves as a practical guideline for choosing and tuning SAT solvers for the many-objective optimal software product selection problem.

Stochastic Local Search for Satisfiability Modulo Theories

Satisfiability Modulo Theories (SMT) is essential for many practical applications, e.g., in hard- and software verification, and increasingly also in other scientific areas like computational biology. A large number of applications in these areas benefit from bit-precise reasoning over finite-domain variables. Current approaches in this area translate a formula over bit-vectors to an equisatisfiable propositional formula, which is then given to a SAT solver. In this paper, we present a novel stochastic local search (SLS) algorithm to solve SMT problems, especially those in the theory of bit-vectors, directly on the theory level. We explain how several successful techniques used in modern SLS solvers for SAT can be lifted to the SMT level. Experimental results show that our approach can compete with state-of-the-art bit-vector solvers on many practical instances and, sometimes, outperform existing solvers. This offers interesting possibilities in combining our approach with existing techniques, and, moreover, new insights into the importance of exploiting problem structure in SLS solvers for SAT. Our approach is modular and, therefore, extensible to support other theories, potentially allowing SLS to become part of the more general SMT framework.

An effective variable selection heuristic in SLS for weighted Max-2-SAT

Stochastic local search (SLS) is an appealing method for solving the maximum satisfiability (Max-SAT) problem. This paper proposes a new variable selection heuristic for Max-SAT local search algorithms, which works particularly well for weighted Max-2-SAT instances. Evolving from the recent configuration checking strategy, this new heuristic works in three levels and is called CCTriplex. According to the CCTriplex heuristic, a variable that is both decreasing and configuration changed has the higher priority to be flipped than a decreasing variable, which in turn has the higher priority than a configuration changed variable. The CCTriplex heuristic is used to develop a new SLS algorithm for weighted Max-2-SAT called CCMaxSAT. We evaluate CCMaxSAT on random benchmarks with different densities, and the hand crafted Frb benchmark, as well as weighted Max-2-SAT instances encoded from MaxCut, MaxClique and sports scheduling problems. Compared with the state-of-the-art SLS solver for weighted Max-2-SAT called ITS and the best SLS solver in Max-SAT Evaluation 2012 namely ubcsat-IRoTS, as well as the famous complete solver wMaxSATz, our algorithm CCMaxSAT shows rather good performance on all the benchmarks.

Scoring Functions Based on Second Level Score for k-SAT with Long Clauses

It is widely acknowledged that stochastic local search (SLS) algorithms can efficiently find models for satisfiable instances of the satisfiability (SAT) problem, especially for random k-SAT instances. However, compared to random 3-SAT instances where SLS algorithms have shown great success, random k-SAT instances with long clauses remain very difficult. Recently, the notion of second level score, denoted as "score_2", was proposed for improving SLS algorithms on long-clause SAT instances, and was first used in the powerful CCASat solver as a tie breaker. In this paper, we propose three new scoring functions based on score_2. Despite their simplicity, these functions are very effective for solving random k-SAT with long clauses. The first function combines score and score_2, and the second one additionally integrates the diversification property "age". These two functions are used in developing a new SLS algorithm called CScoreSAT. Experimental results on large random 5-SAT and 7-SAT instances near phase transition show that CScoreSAT significantly outperforms previous SLS solvers. However, CScoreSAT cannot rival its competitors on random k-SAT instances at phase transition. We improve CScoreSAT for such instances by another scoring function which combines score_2 with age. The resulting algorithm HScoreSAT exhibits state-of-the-art performance on random k-SAT (k>3) instances at phase transition. We also study the computation of score_2, including its implementation and computational complexity.

More efficient two-mode stochastic local search for random 3-satisfiability

Stochastic local search (SLS) is a popular paradigm in incomplete solving for the Boolean satisfiability problem (SAT). Most SLS solvers for SAT switch between two modes, i.e., the greedy (intensif...

Double Configuration Checking in Stochastic Local Search for Satisfiability

Stochastic local search (SLS) algorithms have shown effectiveness on satisfiable instances of the Boolean satisfiability (SAT) problem. However, their performance is still unsatisfactory on random k-SAT at the phase transition, which is of significance and is one of the empirically hardest distributions of SAT instances. In this paper, we propose a new heuristic called DCCA, which combines two configuration checking (CC) strategies with different definitions of configuration in a novel way. We use the DCCA heuristic to design an efficient SLS solver for SAT dubbed DCCASat. The experiments show that the DCCASat solver significantly outperforms a number of state-of-the-art solvers on extensive random k-SAT benchmarks at the phase transition. Moreover, DCCASat shows good performance on structured benchmarks, and a combination of DCCASat with a complete solver achieves state-of-the-art performance on structured benchmarks.

On Improving Local Search for Unsatisfiability

Stochastic local search (SLS) has been an active field of research in the last few years, with new techniques and procedures being developed at an astonishing rate. SLS has been traditionally associated with satisfiability solving, that is, finding a solution for a given problem instance, as its intrinsic nature does not address unsatisfiable problems. Unsatisfiable instances were therefore commonly solved using backtrack search solvers. For this reason, in the late 90s Selman, Kautz and McAllester proposed a challenge to use local search instead to prove unsatisfiability. More recently, two SLS solvers - Ranger and Gunsat - have been developed, which are able to prove unsatisfiability albeit being SLS solvers. In this paper, we first compare Ranger with Gunsat and then propose to improve Ranger performance using some of Gunsat's techniques, namely unit propagation look-ahead and extended resolution.