Non-deterministic polynomial time (commonly termed`NP-complete') problems are relevant to many computational tasks of practical interestÐsuch as the`travelling salesman problem'Ðbut are dif®cult to solve: the computing time grows exponentially with problem size in the worst case. It has recently been shown that these problems exhibit`phase boundaries', across which dramatic changes occur in the computational dif®culty and solution characterÐthe problems become easier to solve away from the boundary. Here we report an analytic solution and experimental investigation of the phase transition in K-satis®ability, an archetypal NP-complete problem. Depending on the input parameters, the computing time may grow exponentially or polynomially with problem size; in the former case, we observe a discontinuous transition, whereas in the latter case a continuous (second-order) transition is found. The nature of these transitions may explain the differing computational costs, and suggests directions for improving the ef®ciency of search algorithms. Similar types of transition should occur in other combinatorial problems and in glassy or granular materials, thereby strengthening the link between computational models and properties of physical systems.Many computational tasks of practical interest are surprisingly dif®cult to solve even using the fastest available machines. Such problems, found for example in planning, scheduling, machine learning, hardware design, and computational biology, generally belong to the class of NP-complete problems 1±3 . NP stands for`nondeterministic polynomial time', which denotes an abstract computational model with a rather technical de®nition. Intuitively speaking, this class of computational tasks consists of problems for which a potential solution can be checked ef®ciently for correctness, yet ®nding such a solution appears to require exponential time in the worst case. A good analogy can be drawn from mathematics: proving open conjectures in mathematics is extremely dif®cult, but verifying any given proof (or solution) is generally relatively straightforward.The class of NP-complete problems lies at the foundations of the theory of computational complexity in modern computer science. Literally thousands of computational problems have been shown to be NP-complete. The completeness property of NPcomplete problems means that if an ef®cient algorithm for solving just one of these problems could be found, one would immediately have an ef®cient algorithm for all NP-complete problems. However, a fundamental conjecture of modern complexity theory is that no such ef®cient algorithm exists.Although NP-complete problems are believed to require exponential time to solve in the worst case, the typical-case behaviour is dif®cult to characterize. Yet, such typical-case properties are most relevant in practical applications. Fu and Anderson 4 ®rst conjectured a deep connection between NP-complete problems and models studied in statistical physics 5 . More recently, it was discovered (refs 6±9) that NP-compl...

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Determining the satisfiability of randomly generated Boolean expressions with k variables per clause is a popular test for the performance of search algorithms in artificial intelligence and computer science. It is known that for k = 2, formulas are almost always satisfiable when the ratio of clauses to variables is less than 1; for ratios larger than 1, the formulas are almost never satisfiable. Similar sharp threshold behavior is observed for higher values of k. Finite-size scaling, a method from statistical physics, can be used to characterize size-dependent effects near the threshold. A relationship can be drawn between thresholds and computational complexity.

It has recently been shown that local search is surprisingly good at nding satisfying assignments for certain classes of CNF formulas 24]. In this paper we demonstrate that the power of local search for satis ability testing can be further enhanced by employing a new strategy, called \mixed random walk", for escaping from local minima. We present experimental results showing how this strategy allows us to handle formulas that are substantially larger than those that can be solved with basic local search. We also present a detailed comparison of our random walk strategy with simulated annealing. Our results show that mixed random walk is the superior strategy on several classes of computationally di cult problem instances. Finally, we present results demonstrating the e ectiveness of local search with walk for solving circuit synthesis and diagnosis problems.

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We are interested in tracking changes in large-scale data by periodically creating an agglomerative clustering and examining the evolution of clusters (communities) over time. We examine a large real-world data set: the NEC CiteSeer database, a linked network of >250,000 papers. Tracking changes over time requires a clustering algorithm that produces clusters stable under small perturbations of the input data. However, small perturbations of the CiteSeer data lead to significant changes to most of the clusters. One reason for this is that the order in which papers within communities are combined is somewhat arbitrary. However, certain subsets of papers, called natural communities, correspond to real structure in the CiteSeer database and thus appear in any clustering. By identifying the subset of clusters that remain stable under multiple clustering runs, we get the set of natural communities that we can track over time. We demonstrate that such natural communities allow us to identify emerging communities and track temporal changes in the underlying structure of our network data. E mergent properties of large linked networks have recently become the focus of intense study. This research is driven by the increasing complexity and importance of large networks, such as the World Wide Web, the electricity grid, and large social networks that capture relationships between individuals. Realworld networks generally exhibit properties that lie somewhere in-between those of highly structured networks and purely random ones (1-4). So far, most research has focused on using static properties, such as the connectivity of the nodes in the network and the average distance between two nodes, to explain the complex structure. However, these networks generally evolve over time and so temporal characteristics are a key source of interest. Our goal in this paper is to provide techniques for the study of the evolution of large linked networks.In our approach, we use agglomerative clusterings of the linked network. By clustering the network at different points in time, we study its temporal evolution. This approach places a new burden on the underlying clustering method. Clustering methods can be surprisingly sensitive to minor changes of the input data. For obtaining a static view of the higher-level structure of the data, such instabilities may be acceptable because the resulting hierarchy often already reveals interesting structure. However, in tracking changes over time, we need to be able to find corresponding communities in clusterings taken from the data at different points in time. If the clusterings are very sensitive to small perturbations of the input data, distinguishing between ''real'' changes versus ''accidental'' changes in the higher-level structure becomes difficult, if not impossible. In the clusterings of our linked network data, we found there are a large number of relatively random clusters that do not correspond to real community structures. These random clusters obscure the real temporal changes. Fortunately...

Computational efficiency is a central concern in the design of knowledge representation systems. In order to obtain efficient systems, it has been suggested that one should limit the form of the statements in the knowledge base or use an incomplete inference mechanism. The former approach is often too restrictive for practical applications, whereas the latter leads to uncertainty about exactly what can and cannot be inferred from the knowledge base. We present a third alternative, in which knowledge given in a general representation language is translated (compiled) into a tractable form—allowing for efficient subsequent query answering.We show how propositional logical theories can be compiled into Horn theories that approximate the original information. The approximations bound the original theory from below and above in terms of logical strength. The procedures are extended to other tractable languages (for example, binary clauses) and to the first-order case. Finally, we demonstrate the generality of our approach by compiling concept descriptions in a general frame-based language into a tractable form.