Empirical investigation of stochastic local search for maximum satisfiability

The maximum satisfiability (MAX-SAT) problem is an important NP-hard problem in theory, and has a broad range of applications in practice. Stochastic local search (SLS) is becoming an increasingly popular method for solving MAX-SAT. Recently, a powerful SLS algorithm called CCLS shows efficiency on solving random and crafted MAX-SAT instances. However, the performance of CCLS on solving industrial MAX-SAT instances lags far behind. In this paper, we focus on experimentally analyzing the performance of SLS algorithms for solving industrial MAXSAT instances. First, we conduct experiments to analyze why CCLS performs poor on industrial instances. Then we propose a new strategy called additive BMS (Best from Multiple Selections) to ease the serious issue. By integrating CCLS and additive BMS, we develop a new SLS algorithm for MAXSAT called CCABMS, and related experiments indicate the efficiency of CCABMS. Also, we experimentally analyze the effectiveness of initialization methods on SLS algorithms for MAX-SAT, and combine an effective initialization method with CCABMS, resulting in an enhanced algorithm. Experimental results show that our enhanced algorithm performs better than its state-of-the-art SLS competitors on a large number of industrial MAX-SAT instances.     

Quantified maximum satisfiability

In recent years, there have been significant improvements in algorithms both for Quantified Boolean Formulas (QBF) and for Maximum Satisfiability (MaxSAT). This paper studies an optimization extension of QBF and considers the problem in a quantified MaxSAT setting. More precisely, the general QMaxSAT problem is defined for QBFs with a set of soft clausal constraints and consists in finding the largest subset of the soft constraints such that the remaining QBF is true. Two approaches are investigated. One is based on relaxing the soft clauses and performing an iterative search on the cost function. The other approach, which is the main contribution of the paper, is inspired by recent work on MaxSAT, and exploits the iterative identification of unsatisfiable cores. The paper investigates the application of these approaches to the two concrete problems of computing smallest minimal unsatisfiable subformulas (SMUS) and smallest minimal equivalent subformulas (SMES), decision versions of which are well-known problems in the second level of the polynomial hierarchy. Experimental results, obtained on representative problem instances, indicate that the core-guided approach for the SMUS and SMES problems outperforms the use of iterative search over the values of the cost function. More significantly, the core-guided approach to SMUS also outperforms the state-of-the-art SMUS extractor Digger.     

Algorithms for the maximum satisfiability problem

Old and new algorithms for the Maximum Satisfiability problem are studied. We first summarize the different heuristics previously proposed, i.e., the approximation algorithms of Johnson and of Lieberherr for the general Maximum Satisfiability problem, and the heuristics of Lieberherr and Specker, Poljak and Turzik for the Maximum 2-Satisfiability problem. We then consider two recent local search algorithmic schemes, the Simulated Annealing method of Kirkpatrick, Gelatt and Vecchi and the Steepest Ascent Mildest Descent method, and adapt them to the Maximum Satisfiability problem. The resulting algorithms, which avoid being blocked as soon as a local optimum has been found, are shown empirically to be more efficient than the heuristics previously proposed in the literature.     

An effective heuristic algorithm for the maximum satisfiability problem

Stochastic local search algorithms (SLS) have been increasingly applied to approximate solutions of the weighted maximum satisfiability problem (MAXSAT), a model for solutions of major problems in AI and combinatorial optimization. While MAXSAT instances have generally a strong intrinsic dependency between their variables, most of SLS algorithms start the search process with a random initial solution where the value of each variable is generated independently with the same uniform distribution. In this paper, we propose a new SLS algorithm for MAXSAT based on an unconventional distribution known as the Bose-Einstein distribution in quantum physics. It provides a stochastic initialization scheme to an efficient and very simple heuristic inspired by the co-evolution process of natural species and called Extremal Optimization (EO). This heuristic was introduced for finding high quality solutions to hard optimization problems such as colouring and partitioning. We examine the effectiveness of the resulting algorithm by computational experiments on a large set of test instances and compare it with some of the most powerful existing algorithms. Our results are remarkable and show that this approach is appropriate for this class of problems.     

An Algorithm Based on Tabu Search for Satisfiability Problem

In this paper,a computationally effective algorithm based on tabu search for solving the satisfiability problem(TSSAT)is proposed.Some novel and efficient heuristic strategies for generating candidate neighborhood of the curred assignment and selecting varibables to be flipped are presented. Especially,the aspiration criterion and tabu list tructure of TSSAT are different from those of traditional tabu search.Computational experiments on a class of problem insteances show that,TSSAT,in a reasonable amount of computer time ,yields better results than Novelty which is currently among the fastest known.Therefore TSSAT is feasible and effective.     

Automated discovery of local search heuristics for satisfiability testing

The development of successful metaheuristic algorithms such as local search for a difficult problem such as satisfiability testing (SAT) is a challenging task. We investigate an evolutionary approach to automating the discovery of new local search heuristics for SAT. We show that several well-known SAT local search algorithms such as Walksat and Novelty are composite heuristics that are derived from novel combinations of a set of building blocks. Based on this observation, we developed CLASS, a genetic programming system that uses a simple composition operator to automatically discover SAT local search heuristics. New heuristics discovered by CLASS are shown to be competitive with the best Walksat variants, including Novelty+. Evolutionary algorithms have previously been applied to directly evolve a solution for a particular SAT instance. We show that the heuristics discovered by CLASS are also competitive with these previous, direct evolutionary approaches for SAT. We also analyze the local search behavior of the learned heuristics using the depth, mobility, and coverage metrics proposed by Schuurmans and Southey.     

GASAT: a genetic local search algorithm for the satisfiability problem

This paper presents GASAT, a hybrid algorithm for the satisfiability problem (SAT). The main feature of GASAT is that it includes a recombination stage based on a specific crossover and a tabu search stage. We have conducted experiments to evaluate the different components of GASAT and to compare its overall performance with state-of-the-art SAT algorithms. These experiments show that GASAT provides very competitive results.     

Planning as satisfiability: parallel plans and algorithms for plan search

We address two aspects of constructing plans efficiently by means of satisfiability testing: efficient encoding of the problem of existence of plans of a given number t of time points in the propositional logic and strategies for finding plans, given these formulae for different values of t.For the first problem we consider three semantics for plans with parallel operator application in order to make the search for plans more efficient. The standard semantics requires that parallel operators are independent and can therefore be executed in any order. We consider a more relaxed definition of parallel plans which was first proposed by Dimopoulos et al., as well as a normal form for parallel plans that requires every operator to be executed as early as possible. We formalize the semantics of parallel plans emerging in this setting and present translations of these semantics into the propositional logic. The sizes of the translations are asymptotically optimal. Each of the semantics is constructed in such a way that there is a plan following the semantics exactly when there is a sequential plan, and moreover, the existence of a parallel plan implies the existence of a sequential plan with as many operators as in the parallel one.For the second problem we consider strategies based on testing the satisfiability of several formulae representing plans of n time steps for several values of n concurrently by several processes. We show that big efficiency gains can be obtained in comparison to the standard strategy of sequentially testing the satisfiability of formulae for an increasing number of time steps.     

Parallel search for LTL violations

Recent advances in parallel model checking for liveness properties achieve significant capacity increases over sequential model checkers. However, the capacity of parallel model checkers is in turn limited by available aggregate memory and network bandwidth. We propose a new parallel algorithm that sacrifices complete coverage for increased capacity to find errors. The algorithm, called BEE (for bee-based error exploration), uses coordinated depth-bounded random walks to reduce memory and bandwidth demands. A unique advantage of BEE is that it is well suited for use on clusters of nondedicated workstations.     

Parallel game-tree search

The design issues affecting a parallel implementation of the alpha-beta search algorithm are discussed with emphasis on a tree decomposition scheme that is intended for use on well ordered trees. In particular, the principal variation splitting method has been implemented, and experimental results are presented which show how such refinements as progressive deepening, narrow window searching, and the use of memory tables affect the performance of multiprocessor based chess playing programs. When dealing with parallel processing systems, communication delays are perhaps the greatest source of lost time. Therefore, an implementation of our tree decomposition based algorithm is presented, one that operates with a modest amount of message passing within a network of processors. Since our system has low search overhead, the principal basis for comparison is the communication overhead, which in turn is shown to have two components.     

Infra-chromatic bound for exact maximum clique search

Many efficient exact branch and bound maximum clique solvers use approximate coloring to compute an upper bound on the clique number for every subproblem. This technique reasonably promises tight bounds on average, but never tighter than the chromatic number of the graph.Li and Quan, 2010, AAAI Conference, p. 128–133 describe a way to compute even tighter bounds by reducing each colored subproblem to maximum satisfiability problem (MaxSAT). Moreover they show empirically that the new bounds obtained may be lower than the chromatic number.Based on this idea this paper shows an efficient way to compute related “infra-chromatic” upper bounds without an explicit MaxSAT encoding. The reported results show some of the best times for a stand-alone computer over a number of instances from standard benchmarks.     

Parallel binary search

Two arrays of numbers sorted in nondecreasing order are given: an array A of size n and an array B of size m, where n<m. It is required to determine, for every element of A, the smallest element of B (if one exists) that is larger than or equal to it. It is shown how to solve this problem on the EREW PRAM (exclusive-read exclusive-write parallel random-access machine) in O(logm logn/log log m) time using n processors. The solution is then extended to the case in which fewer than n processors are available. This yields an EREW PRAM algorithm for the problem whose cost is O(n log m, which is O(m)) for n⩽m/log m. It is shown how the solution obtained leads to an improved parallel merging algorithm     

Adding cardinality constraints to integer programs with applications to maximum satisfiability

Max-SAT-CC is the following optimization problem: Given a formula in CNF and a bound k, find an assignment with at most k variables being set to true that maximizes the number of satisfied clauses among all such assignments. If each clause is restricted to have at most ? literals, we obtain the problem Max-?SAT-CC. Sviridenko [Algorithmica 30 (3) (2001) 398-405] designed a (1−e⁻¹)-approximation algorithm for Max-SAT-CC. This result is tight unless P=NP [U. Feige, J. ACM 45 (4) (1998) 634-652]. Sviridenko asked if it is possible to achieve a better approximation ratio in the case of Max-?SAT-CC. We answer this question in the affirmative by presenting a randomized approximation algorithm whose approximation ratio is . To do this, we develop a general technique for adding a cardinality constraint to certain integer programs. Our algorithm can be derandomized using pairwise independent random variables with small probability space.     

Frequency-driven tabu search for the maximum s-plex problem

The maximum s-plex problem is an important model for social network analysis and other studies. In this study, we present an effective frequency-driven multi-neighborhood tabu search algorithm (FD-TS) to solve the problem on very large networks. The proposed FD-TS algorithm relies on two transformation operators (Add and Swap) to locate high-quality solutions, and a frequency-driven perturbation operator (Press) to escape and search beyond the identified local optimum traps. We report computational results for 47 massive real-life (sparse) graphs from the SNAP Collection and the 10th DIMACS Challenge, as well as 52 (dense) graphs from the 2nd DIMACS Challenge (results for 48 more graphs are also provided in the Appendix). We demonstrate the effectiveness of our approach by presenting comparisons with the current best-performing algorithms.     

Comparing the performance of the genetic and local search algorithms for solving the satisfiability problems

The satisfiability problem (SAT), as one of the six basic core NP-complete problems, has been the deserving object of many studies in the last two decades [1]. Stochastic local search and genetic algorithms are two current state-of-the-art techniques for solving the SATs. GASAT [2,1] and SAT-WAGA [4] are two current state-of-the-art genetic algorithms for solving SATs. Beside, the discrete lagrange-multiplier (DLM) [8–10] and the exponentiated subgradient algorithms (ESG) [12] are the current state-of-the-art local search algorithms for solving SATs.In this paper, we compare DLM and ESG with GASAT and SAT-WAGA. We show that the performance of the local search-based algorithms: DLM and ESG is better than the performance of the genetic-based algorithms: GASAT and SAT-WAGA. We further analyze the results of the comparisons. We hope these comparisons give light to the researchers while researching in genetic and local search algorithms, help understanding of the reasons for these algorithms’ performance and behaviour and to the new researchers who are in the process of choosing research direction.     

求解最大割问题的多启动禁忌搜索算法

为了增强局部搜索算法在求解最大割问题上的寻优能力,提高解质量,提出了一种多启动禁忌搜索（MSTS）算法。算法主要包括两个重要组件:一是用于搜索高质量局部优化解的禁忌搜索算法;二是具有全局搜索能力的重启策略。算法首先通过禁忌搜索组件获取局部优化解;然后应用设计的重启策略重新生成初始解并重启禁忌搜索过程。重启策略基于随机贪心的思想,综合利用了“构造”和“扰动”这两种方法生成新的起始解,来逃离局部最优的陷阱从而找到更高优度的解。采用了国际文献中公认的21个算例作为本算法的测试实验集并进行实算, 并与多个先进算法进行比较,MSTS算法在18个算例上得到最好解值,高于其他对比算法。实验结果表明,MSTS算法具有更强的寻优能力和更高的解质量。     

Branching rules for satisfiability

Recent experience suggests that branching algorithms are among the most attractive for solving propositional satisfiability problems. A key factor in their success is the rule they use to decide on which variable to branch next. We attempt to explain and improve the performance of branching rules with an empirical model-building approach. One model is based on the rationale given for the Jeroslow-Wang rule, variations of which have performed well in recent work. The model is refuted by carefully designed computational experiments. A second model explains the success of the Jeroslow-Wang rule, makes other predictions confirmed by experiment, and leads to the design of branching rules that are clearly superior to Jeroslow-Wang.GSIA Working Paper 1994-09. The first author is partially supported by ONR grant N00014-92-J-1028. The authors wish to thank Ajai Kapoor for assistance in computational p667 testing and statistical analysis.     

An effective local search for the maximum clique problem

We propose a variable depth search based algorithm, called k-opt local search (KLS), for the maximum clique problem. KLS efficiently explores the k-opt neighborhood defined as the set of neighbors that can be obtained by a sequence of several add and drop moves that are adaptively changed in the feasible search space. Computational results on DIMACS benchmark graphs indicate that KLS is capable of finding considerably satisfactory cliques with reasonable running times in comparison with those of state-of-the-art metaheuristics.     

Improved initial vertex ordering for exact maximum clique search

This paper describes a new initial vertexordering procedure NEW_SORT designed to enhance approximate-colour exact algorithms for the maximum clique problem (MCP). NEW_SORT considers two different vertex orderings: degree and colour-based. The degree-based vertex ordering describes an improvement over a well-known vertex ordering used by exact solvers. Moreover, colour-based vertex orderings for the MCP have been traditionally considered suboptimal with respect to degree-based ones. NEW_SORT chooses the “best” of the two orderings according to a new evaluation function. The reported experiments on graphs taken from public datasets show that a leading exact solver using NEW_SORT —and further enhanced with a strong initial solution— can improve its performance very significantly (sometimes even exponentially).