Many different approaches have been applied to constraint satisfaction. These range from complete backtracking algorithms to sophisticated distributed configurations. However, most research effort in the field of constraint satisfaction algorithms has concentrated on the use of a single algorithm for solving all problems. At the same time, a consensus appears to have developed to the effect that it is unlikely that any single algorithm is always the best choice for all classes of problem. In this paper we argue that an adaptive approach should play an important part in constraint satisfaction. This approach relaxes the commitment to using a single algorithm once search commences. As a result, we claim that it is possible to undertake a more focused approach to problem solving, allowing for the correction of bad algorithm choices and for capitalising on opportunities for gain by dynamically changing to more suitable candidates.

Stochastic local search (SLS) algorithms for the propositional satisfiability problem (SAT) have been successfully applied to solve suitably encoded search problems from various domains. One drawback of these algorithms is that they are usually incomplete. We refine the notion of incompleteness for stochastic decision algorithms by introducing the notion of "probabilistic asymptotic completeness" (PAC) and prove for a number of well-known SLS algorithms whether or not they have this property. We also give evidence for the practical impact of the PAC property and show how to achieve the PAC property and significantly improved performance in practice for some of the most powerful SLS algorithms for SAT, using a simple and general technique called "random walk extension".

A major difficulty in evaluating incomplete local search style algorithms for constraint satisfaction problems is the need for a source of hard problem instances that are guaranteed to be satisfiable. A standard approach to evaluate incomplete search methods has been to use a general problem generator and a complete search method to filter out the unsatisfiable instances. Unfortunately, this approach cannot be used to create problem instances that are beyond the reach of complete search methods. So far, it has proven to be surprisingly difficult to develop a direct generator for satisfiable instances only. In this paper, we propose a generator that only outputs satisfiable problem instances. We also show how one can finely control the hardness of the satisfiable instances by establishing a connection between problem hardness and a new kind of phase transition phenomenon in the space of problem instances. Finally, we use our problem distribution to show the easy-hard-easy pattern in search complexity for local search procedures, analogous to the previously reported pattern for complete search methods.

We wish to bring to the attention of the OR community the phenomenon of phase transitions in randomly generated problems. These are of considerable practical use for benchmarking algorithms. They also offer insight into problem hardness and algorithm performance. Whilst phase transition experiments are frequently performed by AI researchers, such experiments do not appear to be in common use in the OR community. To illustrate the value of such experiments, we examine a typical OR problem, the traveling salesman problem. We report in detail many features of the phase transition in this problem, and show how some of these features are also seen in real problems. Acknowledgements The second author is supported by a HCM Postdoctoral Fellowship. We thank Iain Buchanan for comments on a draft of this paper, and Alan Bundy, and the members of the Mathematical Reasoning Group in Edinburgh for their constructive comments and many CPU cycles donated to these and other experiments from SERC grant GR/H/23610. We also thank the MRG group at Trento and the Department of Computer Science at the University of Strathclyde for additional CPU cycles. Finally, we thank Robert Craig for providing us with his code. 1

Local search algorithms for combinatorial search problems frequently encounter a sequence of states in which it is impossible to improve the value of the objective function; moves through these regions, called plateau moves, dominate the time spent in local search. We analyze and characterize plateaus for three different classes of randomly generated Boolean Satisfiability problems. We identify several interesting features of plateaus that impact the performance of local search algorithms. We show that local minima tend to be small but occasionally may be very large. We also show that local minima can be escaped without unsatisfying a large number of clauses, but that systematically searching for an escape route may be computationally expensive if the local minimum is large. We show that plateaus with exits, called benches, tend to be much larger than minima, and that some benches have very few exit states which local search can use to escape. We show that the solutions (i.e., global m...

Local search algorithms are among the standard methods for solving hard combinatorial problems from various areas of Artificial Intelligence and Operations Research. For SAT, some of the most successful and powerful algorithms are based on stochastic local search and in the past 10 years a large number of such algorithms have been proposed and investigated. In this article, we focus on two particularly well-known families of local search algorithms for SAT, the GSAT and WalkSAT architectures. We present a detailed comparative analysis of these algorithms' performance using a benchmark set which contains instances from randomised distributions as well as SAT-encoded problems from various domains. We also investigate the robustness of the observed performance characteristics as algorithm-dependent and problem-dependent parameters are changed. Our empirical analysis gives a very detailed picture of the algorithms' performance for various domains of SAT problems; it also reveals a fundamental weakness in some of the best-performing algorithms and shows how this can be overcome.

Stochastic search algorithms are among the most sucessful approaches for solving hard combinatorial problems. A large class of stochastic search approaches can be cast into the framework of Las Vegas Algorithms (LVAs). As the run-time behavior of LVAs is characterized by random variables, the detailed knowledge of run-time distributions provides important information for the analysis of these algorithms. In this paper we propose a novel methodology for evaluating the performance of LVAs, based on the identification of empirical run-time distributions. We exemplify our approach by applying it to Stochastic Local Search (SLS) algorithms for the satisfiability problem (SAT) in propositional logic. We point out pitfalls arising from the use of improper empirical methods and discuss the benefits of the proposed methodology for evaluating and comparing LVAs.

Several local search algorithms for propositional satisfiability have been proposed which can solve hard random problems beyond the range of conventional backtracking procedures. In this paper, we explore the impact of focusing search in these procedures on the "unsatisfied variables"; that is, those variables which appear in clauses which are not yet satisfied. For random problems, we show that such a focus reduces the sensitivity to input parameters. We also observe a simple scaling law in performance. For non-random problems, we show that whilst this focus can improve performance, many problems remain difficult. We speculate that such problems will remain hard for local search unless constraint propagation techniques can be combined with hill-climbing. 1 Introduction Local search is often surprisingly effective as a semi-decision procedure for many NPhard problems. For example, Gsat, a greedy random hill-climbing procedure for propositional satisfiability (or SAT) is very good at ...

. There has been considerable research interest into the solubility phase transition, and its effect on search cost for backtracking algorithms. In this paper we show that a similar easy-hard-easy pattern occurs for local search, with search cost peaking at the phase transition. This is despite problems beyond the phase transition having fewer solutions, which intuitively should make the problems harder to solve. We examine the relationship between search cost and number of solutions at different points across the phase transition, for three different local search procedures, across two problem classes (CSP and SAT). Our findings show that there is a significant correlation, which changes as we move through the phase transition. Keywords: computational complexity, constraint satisfaction, propositional satisfiability, search 1 Introduction Local search has been proposed as a good candidate for solving the "hard" but soluble problems that turn up at the phase transition in solubility f...

Stochastic local search (SLS) algorithms have been successfully applied to hard combinatorial problems from different domains. Due to their inherent randomness, the run-time behaviour of these algorithms is characterised by a random variable. The detailed knowledge of the run-time distribution provides important information about the behaviour of SLS algorithms. In this paper we investigate the empirical run-time distributions for Walksat, one of the most powerful SLS algorithms for the Propositional Satisfiability Problem (SAT). Using statistical analysis techniques, we show that on hard Random-3-SAT problems, Walksat's run-time behaviour can be characterised by exponential distributions. This characterisation can be generalised to various SLS algorithms for SAT and to encoded problems from other domains. This result also has a number of consequences which are of theoretical as well as practical interest. One of these is the fact that these algorithms can be easily parallelised such that optimal speed-up is achieved for hard problem instances.