Improving configuration checking for satisfiable random <Emphasis Type="Italic">k</Emphasis>-SAT instances

Local search algorithms based on the Configuration Checking (CC) strategy have been shown to be efficient in solving satisfiable random k-SAT instances. The purpose of the CC strategy is to avoid the cycling problem, which corresponds to revisiting already flipped variables too soon. It is done by considering the neighborhood of the formula variables. In this paper, we propose to improve the CC strategy on the basis of an empirical study of a powerful algorithm using this strategy. The first improvement introduces a new and simple criterion, which refines the selection of the variables to flip for the 3-SAT instances. The second improvement is achieved by using the powerful local search algorithm Novelty with the adaptive noise setting. This algorithm enhances the efficiency of the intensification and diversification phases when solving k-SAT instances with k ≥ 4. We name the resulting local search algorithm Ncca+ and show its effectiveness when treating satisfiable random k-SAT instances issued from the SAT Challenge 2012. Ncca+ won the bronze medal of the random SAT track of the SAT Competition 2013.     

Beyond the structure of SAT formulas

Nowadays, many real-world problems are encoded into SAT instances and efficiently solved by modern SAT solvers. These solvers, usually known as Conflict-Driven Clause Learning (CDCL) SAT solvers, include a variety of sophisticated techniques, such as clause learning, lazy data structures, conflict-based adaptive branching heuristics, or random restarts, among others. However, the reasons of their efficiency in solving real-world, or industrial, SAT instances are still unknown. The common wisdom in the SAT community is that these technique exploit some hidden structure of real-world problems.In this thesis, we characterize some important features of the underlying structure of industrial SAT instances. Namely, they are the community structure and the self-similar structure. We observe that most industrial SAT formulas, viewed as graphs, have these two properties. This means that (i) in a graph with a clear community structure, i.e. having high modularity, we can find a partition of its nodes into communities such that most edges connect nodes of the same community; and (ii) in a graph with a self-similar pattern, i.e. being fractal, its shape is kept after re-scalings, i.e., grouping sets of nodes into a single node. We also analyze how these structures are affected by the effects of CDCL techniques during the search.Using the previous structural studies, we propose three applications. First, we face the problem of generating pseudo-industrial random SAT instances using the notion of modularity. Our model generates instances similar to (classical) random SAT formulas when the modularity is low, but when this value is high, our model is also adequate to model realistic pseudo-industrial problems. Second, we propose a method based on the community structure of the instance to detect relevant learnt clauses. Our technique augments the original instance with this set of relevant clauses, and this results into an overall improvement of the efficiency of several state-of-the-art CDCL SAT solvers. Finally, we analyze the classification of industrial SAT instances into families using the previously analyzed structure features, and we compare them to other classifiers commonly used in portfolio SAT approaches.In summary, this dissertation extends the understandings of the structure of SAT instances, with the aim of better explaining the success of CDCL techniques and possibly improve them, and propose a number of applications based on this analysis of the underlying structure of SAT formulas.     

SAT问题中局部搜索法的改进

局部搜索方法在求解SAT问题的高效率使其成为一研究热点．提出用初始概率的方法对局部搜索算法中变量的初始随机指派进行适当的约束．使在局部搜索的开始阶段,可满足的子句数大大增加,减少了翻转的次数,加快了求解的速度．用该方法对目前的一些重要的SAT问题的局部搜索算法(如WSAT,TSAT,NSAT,SDF等)进行改进,通过对不同规模的随机3-SAT问题的实例和一些不同规模的结构性SAT问题的实例,以及利用相变现象构造的难解SAT实例测试表明,改进后的这些局部搜索算法的求解效率有了很大的提高．该方法对其他局部搜索法的改进具有参考价值。     

A comparative runtime analysis of heuristic algorithms for satisfiability problems

The satisfiability problem is a basic core NP-complete problem. In recent years, a lot of heuristic algorithms have been developed to solve this problem, and many experiments have evaluated and compared the performance of different heuristic algorithms. However, rigorous theoretical analysis and comparison are rare. This paper analyzes and compares the expected runtime of three basic heuristic algorithms: RandomWalk, (1+1) EA, and hybrid algorithm. The runtime analysis of these heuristic algorithms on two 2-SAT instances shows that the expected runtime of these heuristic algorithms can be exponential time or polynomial time. Furthermore, these heuristic algorithms have their own advantages and disadvantages in solving different SAT instances. It also demonstrates that the expected runtime upper bound of RandomWalk on arbitrary k-SAT (k?3) is O(n(k−1)), and presents a k-SAT instance that has Θ(n(k−1)) expected runtime bound.     

Regular Random k-SAT: Properties of Balanced Formulas

We consider a model for generating random k-SAT formulas, in which each literal occurs approximately the same number of times in the formula clauses (regular random and k-SAT). Our experimental results show that such regular random k-SAT instances are much harder than the usual uniform random k-SAT problems. This is in agreement with other results that show that more balanced instances of random combinatorial problems are often much more difficult to solve than uniformly random instances, even at phase transition boundaries. There are almost no formal results known for such problem distributions. The balancing constraints add a dependency between variables that complicates a standard analysis. Regular random 3-SAT exhibits a phase transition as a function of the ratio α of clauses to variables. The transition takes place at approximately α = 3.5. We show that for α > 3.78 with high probability (w.h.p.) random regular 3-SAT formulas are unsatisfiable. Specifically, the events hold with high probability if Pr when n → ∞. We also show that the analysis of a greedy algorithm proposed by Kaporis et al. for the uniform 3-SAT model can be adapted for regular random 3-SAT. In particular, we show that for formulas with ratio α < 2.46, a greedy algorithm finds a satisfying assignment with positive probability.     

The satisfiability problem in regular CNF-formulas

 In this paper we deal with the propositional satisfiability (SAT) problem for a kind of multiple-valued clausal forms known as regular CNF-formulas and extend some known results from classical logic to this kind of formulas. We present a Davis–Putnam-style satisfiability checking procedure for regular CNF-formulas equipped with suitable data structures and prove its completeness. Then, we describe a series of experiments for regular random 3-SAT instances. We observe that, for the regular 3-SAT problem with this procedure, there exists a threshold of the ratio of clauses to variables such that (i) the most computationally difficult instances tend to be found near the threshold, (ii) there is a sharp transition from satisfiable to unsatisfiable instances at the threshold and (iii) the value of the threshold increases as the number of truth values considered increases. Instances in the hard part provide benchmarks for the evaluation of regular satisfiability solvers.     

Adaptive memory-based local search for MAX-SAT

Many real world problems, such as circuit designing and planning, can be encoded into the maximum satisfiability problem (MAX-SAT). To solve MAX-SAT, many effective local search heuristic algorithms have been reported in the literature. This paper aims to study how useful information could be gathered during the search history and used to enhance local search heuristic algorithms. For this purpose, we present an adaptive memory-based local search heuristic (denoted by AMLS) for solving MAX-SAT. The AMLS algorithm uses several memory structures to define new rules for selecting the next variable to flip at each step and additional adaptive mechanisms and diversification strategies. The effectiveness and efficiency of our AMLS algorithm is evaluated on a large range of random and structured MAX-SAT and SAT instances, many of which are derived from real world applications. The computational results show that AMLS competes favorably, in terms of several criteria, with four state-of-the-art SAT and MAX-SAT solvers AdaptNovelty+, AdaptG2WSAT_p, IRoTS and RoTS.     

Random 3-SAT: The Plot Thickens

This paper presents an experimental investigation of the following questions: how does the average-case complexity of random 3-SAT, understood as a function of the order (number of variables) for fixed density (ratio of number of clauses to order) instances, depend on the density? Is there a phase transition in which the complexity shifts from polynomial to exponential in the order? Is the transition dependent or independent of the solver? Our experiment design uses three complete SAT solvers embodying different algorithms: GRASP, CPLEX, and CUDD. We observe new phase transitions for all three solvers, where the median running time shifts from polynomial in the order to exponential. The location of the phase transition appears to be solver-dependent. GRASP shifts from polynomial to exponential complexity near the density of 3.8, CPLEX shifts near density 3, while CUDD exhibits this transition between densities of 0.1 and 0.5. This experimental result underscores the dependence between the solver and the complexity phase transition, and challenges the widely held belief that random 3-SAT exhibits a phase transition in computational complexity very close to the crossover point.     

A Parallelization Scheme Based on Work Stealing for a Class of SAT Solvers

Because of the inherent NP-completeness of SAT, many SAT problems currently cannot be solved in a reasonable time. Usually, in order to tackle a new class of SAT problems, new ad hoc algorithms must be designed. Another way to solve a new problem is to use a generic solver and employ parallelism to reduce the solve time. In this paper we propose a parallelization scheme for a class of SAT solvers based on the DPLL procedure. The scheme uses a dynamic load-balancing mechanism based on work-stealing techniques to deal with the irregularity of SAT problems. We parallelize Satz, one of the best generic SAT solvers, with our scheme to obtain a parallel solver called PSatz. The first experimental results on random 3-SAT problems and a set of well-known structured problems show the efficiency of PSatz. PSatz is freely available and runs on any networked workstations under Unix/Linux.     

Heuristic-Based Backtracking Relaxation for Propositional Satisfiability

In recent years backtrack search algorithms for propositional satisfiability (SAT) have been the subject of dramatic improvements. These improvements allowed SAT solvers to successfully solve instances with thousands or tens of thousands of variables. However, many new challenging problem instances are still too hard for current SAT solvers. As a result, further improvements to SAT technology are expected to have key consequences in solving hard real-world instances. This paper introduces a new idea: choosing the backtrack variable using a heuristic approach with the goal of diversifying the regions of the space that are explored during the search. The proposed heuristics are inspired by the heuristics proposed in recent years for the decision branching step of SAT solvers, namely, VSIDS and its improvements. Completeness conditions are established, which guarantee completeness for the new algorithm, as well as for any other incomplete backtracking algorithm. Experimental results on hundreds of instances derived from real-world problems show that the new technique is able to speed SAT solvers, while aborting fewer instances. These results clearly motivate the integration of heuristic backtracking in SAT solvers.     

SAT Solving for Termination Proofs with Recursive Path Orders and Dependency Pairs

This paper introduces a propositional encoding for recursive path orders (RPO), in connection with dependency pairs. Hence, we capture in a uniform setting all common instances of RPO, i.e., lexicographic path orders (LPO), multiset path orders (MPO), and lexicographic path orders with status (LPOS). This facilitates the application of SAT solvers for termination analysis of term rewrite systems (TRSs). We address four main inter-related issues and show how to encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (A) the lexicographic comparison w.r.t. a permutation of the arguments; (B) the multiset extension of a base order; (C) the combined search for a path order together with an argument filter to orient a set of inequalities; and (D) how the choice of the argument filter influences the set of inequalities that have to be oriented (so-called usable rules). We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power.     

随机正则(k,r)-SAT问题的可满足临界

研究k-SAT问题实例中每个变元恰好出现r=2s次,且每个变元对应的正、负文字都出现s次的严格随机正则（k,r）-SAT问题.通过构造一个特殊的独立随机实验,结合一阶矩方法,给出了严格随机正则（k,r）-SAT问题可满足临界值的上界.由于严格正则情形与正则情形的可满足临界值近似相等,因此得到了随机正则（k,r）-SAT问题可满足临界值的新上界.该上界不仅小于当前已有的随机正则（k,r）-SAT问题的可满足临界值上界,而且还小于一般的随机k-SAT问题的可满足临界值.因此,这也从理论上解释了在相变点处的随机正则（k,r）-SAT问题实例通常比在相应相变点处同规模的随机k-SAT问题实例更难满足的原因.最后,数值分析结果验证了所给上界的正确性.     

Improved approximation algorithms for low-density instances of the Minimum Entropy Set Cover Problem

We study the approximability of instances of the minimum entropy set cover problem, parameterized by the average frequency of a random element in the covering sets. We analyze an algorithm combining a greedy approach with another one biased towards large sets. The algorithm is controlled by the percentage of elements to which we apply the biased approach. The optimal parameter choice leads to improved approximation guarantees when average element frequency is less than e.     

On SAT instance classes and a method for reliable performance experiments with SAT solvers

A recent series of experiments with a group of state-of-the-art SAT solvers and several well-defined classes of problem instances reports statistically significant performance variability for the solvers. A systematic analysis of the observed performance data, all openly archived on the Web, reveals distributions which we classify into three broad categories: (1) readily characterized with a simple ²-test, (2) requiring more in-depth analysis by a statistician, (3) incomplete, due to time-out limit reached by specific solvers. The first category includes two well-known distributions: normal and exponential; we use simple first-order criteria to decide the second category and label the distributions as near-normal, near-exponential and heavy-tail. We expect that good models for some if not most of these may be found with parameters that fit either generalized gamma, Weibull, or Pareto distributions. Our experiments show that most SAT solvers exhibit either normal or exponential distribution of execution time (runtime) on many equivalence classes of problem instances. This finding suggests that the basic mathematical framework for these experiments may well be the same as the one used to test the reliability or lifetime of hardware components such as lightbulbs, A/C units, etc. A batch of N replicated hardware components represents an equivalence class of N problem instances in SAT, a controlled operating environment A represents a SAT solver A, and the survival function R ^A (x) (where x represents the lifetime) is the complement of the solvability function S ^A (x)=1–R ^A (x) where x may represent runtime, implications, backtracks, etc. As demonstrated in the paper, a set of unrelated benchmarks or randomly generated SAT instances available today cannot measure the performance of SAT solvers reliably – there is no control on their 'hardness'. However, equivalence class instances as defined in this paper are, in effect, replicated instances of a specific reference instance. The proposed method not only provides a common platform for a systematic study and a reliable improvement of deterministic and stochastic SAT solvers alike but also supports the introduction and validation of new problem instance classes.     

A deterministic (2−2/(k+1))n algorithm for k-SAT based on local search

Local search is widely used for solving the propositional satisfiability problem. Papadimitriou (1991) showed that randomized local search solves 2-SAT in polynomial time. Recently, Schöning (1999) proved that a close algorithm for k-SAT takes time (2−2/k)ⁿ up to a polynomial factor. This is the best known worst-case upper bound for randomized 3-SAT algorithms (cf. also recent preprint by Schuler et al.).We describe a deterministic local search algorithm for k-SAT running in time (2−2/(k+1))ⁿ up to a polynomial factor. The key point of our algorithm is the use of covering codes instead of random choice of initial assignments. Compared to other “weakly exponential” algorithms, our algorithm is technically quite simple. We also describe an improved version of local search. For 3-SAT the improved algorithm runs in time 1.481ⁿ up to a polynomial factor. Our bounds are better than all previous bounds for deterministic k-SAT algorithms.     

A Probabilistic Algorithm for k -SAT Based on Limited Local Search and Restart

A simple probabilistic algorithm for solving the NP-complete problem k -SAT is reconsidered. This algorithm follows a well-known local-search paradigm: randomly guess an initial assignment and then, guided by those clauses that are not satisfied, by successively choosing a random literal from such a clause and changing the corresponding truth value, try to find a satisfying assignment. Papadimitriou [11] introduced this random approach and applied it to the case of 2-SAT, obtaining an expected O(n ² ) time bound. The novelty here is to restart the algorithm after 3n unsuccessful steps of local search. The analysis shows that for any satisfiable k -CNF formula with n variables the expected number of repetitions until a satisfying assignment is found this way is (2? (k-1)/ k) ⁿ . Thus, for 3-SAT the algorithm presented here has a complexity which is within a polynomial factor of (\frac 4 3 ) ⁿ . This is the fastest and also the simplest among those algorithms known up to date for 3-SAT achieving an o(2 ⁿ ) time bound. Also, the analysis is quite simple compared with other such algorithms considered before.     

Solving constraint satisfaction problems with SAT modulo theories

Due to significant advances in SAT technology in the last years, its use for solving constraint satisfaction problems has been gaining wide acceptance. Solvers for satisfiability modulo theories (SMT) generalize SAT solving by adding the ability to handle arithmetic and other theories. Although there are results pointing out the adequacy of SMT solvers for solving CSPs, there are no available tools to extensively explore such adequacy. For this reason, in this paper we introduce a tool for translating FLATZINC (MINIZINC intermediate code) instances of CSPs to the standard SMT-LIB language. We provide extensive performance comparisons between state-of-the-art SMT solvers and most of the available FLATZINC solvers on standard FLATZINC problems. The obtained results suggest that state-of-the-art SMT solvers can be effectively used to solve CSPs.     

警示传播算法收敛的充分条件

信息传播算法求解可满足问题时有惊人的效果,难解区域变窄.然而,因子图带有环的实例,信息传播算法不总有效,常表现为不收敛.对于这种现象,至今缺少系统的理论解释.警示传播（warning propagation,简称WP）算法是一种基础的信息传播算法,对WP算法的收敛性研究是其他信息传播算法收敛性研究的重要基础.在WP算法中,将警示信息的取值从{0,1}松弛为[0,1],利用压缩函数的性质,给出了WP算法收敛的一个充分条件.选取了两组不同规模的随机3-SAT实例进行实验模拟,结果表明：当子句与变元的比值α<1.8时,该判定条件有效.     

Iterative and core-guided MaxSAT solving: A survey and assessment

Maximum Satisfiability (MaxSAT) is an optimization version of SAT, and many real world applications can be naturally encoded as such. Solving MaxSAT is an important problem from both a theoretical and a practical point of view. In recent years, there has been considerable interest in developing efficient algorithms and several families of algorithms have been proposed. This paper overviews recent approaches to handle MaxSAT and presents a survey of MaxSAT algorithms based on iteratively calling a SAT solver which are particularly effective to solve problems arising in industrial settings. First, classic algorithms based on iteratively calling a SAT solver and updating a bound are overviewed. Such algorithms are referred to as iterative MaxSAT algorithms. Then, more sophisticated algorithms that additionally take advantage of unsatisfiable cores are described, which are referred to as core-guided MaxSAT algorithms. Core-guided MaxSAT algorithms use the information provided by unsatisfiable cores to relax clauses on demand and to create simpler constraints. Finally, a comprehensive empirical study on non-random benchmarks is conducted, including not only the surveyed algorithms, but also other state-of-the-art MaxSAT solvers. The results indicate that (i) core-guided MaxSAT algorithms in general abort in less instances than classic solvers based on iteratively calling a SAT solver and that (ii) core-guided MaxSAT algorithms are fairly competitive compared to other approaches.     

Combining finite learning automata with GSAT for the satisfiability problem

A large number of problems that occur in knowledge-representation, learning, very large scale integration technology (VLSI-design), and other areas of artificial intelligence, are essentially satisfiability problems. The satisfiability problem refers to the task of finding a satisfying assignment that makes a Boolean expression evaluate to True. The growing need for more efficient and scalable algorithms has led to the development of a large number of SAT solvers. This paper reports the first approach that combines finite learning automata with the greedy satisfiability algorithm (GSAT). In brief, we introduce a new algorithm that integrates finite learning automata and traditional GSAT used with random walk. Furthermore, we present a detailed comparative analysis of the new algorithm's performance, using a benchmark set containing randomized and structured problems from various domains.