Clustering of solutions in hard satisfiability problems

We study the structure of the solution space and behavior of local search methods on random 3-SAT problems close to the SAT/UNSAT transition. Using the overlap measure of similarity between different solutions found on the same problem instance we show that the solution space is shrinking as a function of alpha. We consider chains of satisfiability problems, where clauses are added sequentially. In each such chain, the overlap distribution is first smooth, and then develops a tiered structure, indicating that the solutions are found in well separated clusters. On chains of not too large instances, all solutions are eventually observed to be in only one small cluster before vanishing. This condensation transition point is estimated to be alpha_c = 4.26. The transition approximately obeys finite-size scaling with an apparent critical exponent of about 1.7. We compare the solutions found by a local heuristic, ASAT, and the Survey Propagation algorithm up to alpha_c.Comment: 8 pages, 9 figure

Optimal Testing for Planted Satisfiability Problems

We study the problem of detecting planted solutions in a random satisfiability formula. Adopting the formalism of hypothesis testing in statistical analysis, we describe the minimax optimal rates of detection. Our analysis relies on the study of the number of satisfying assignments, for which we prove new results. We also address algorithmic issues, and give a computationally efficient test with optimal statistical performance. This result is compared to an average-case hypothesis on the hardness of refuting satisfiability of random formulas

Physics-inspired Performace Evaluation of a Structured Peer-to-Peer Overlay Network

In the majority of structured peer-to-peer overlay networks a graph with a desirable topology is constructed. In most cases, the graph is maintained by a periodic activity performed by each node in the graph to preserve the desirable structure in face of the continuous change of the set of nodes. The interaction of the autonomous periodic activities of the nodes renders the performance analysis of such systems complex and simulation of scales of interest can be prohibitive. Physicists, however, are accustomed to dealing with scale by characterizing a system using intensive variables, i.e. variables that are size independent. The approach has proved its usefulness when applied to satisfiability theory. This work is the first attempt to apply it in the area of distributed systems. The contribution of this paper is two-fold. First, we describe a methodology to be used for analyzing the performance of large scale distributed systems. Second, we show how we applied the methodology to find an intensive variable that describe the characteristic behavior of the Chord overlay network, namely, the ratio of the magnitude of perturbation of the network (joins/failures) to the magnitude of periodic stabilization of the network

Typical-case complexity and the SAT competitions

The aim of this paper is to gather insight into typical-case complexity of the Boolean Satisfiability (SAT) problem by mining the data from the SAT competitions. Specifically, the statistical properties of the SAT benchmarks and their impact on complexity are investigated, as well as connections between different metrics of complexity. While some of the investigated properties and relationships are “folklore” in the SAT community, this study aims at scientifically showing what is true from the folklore and what is not

Random subcube intersection graphs I: cliques and covering

We study random subcube intersection graphs, that is, graphs obtained by selecting a random collection of subcubes of a fixed hypercube $Q_d$ to serve as the vertices of the graph, and setting an edge between a pair of subcubes if their intersection is non-empty. Our motivation for considering such graphs is to model `random compatibility' between vertices in a large network. For both of the models considered in this paper, we determine the thresholds for covering the underlying hypercube $Q_d$ and for the appearance of s-cliques. In addition we pose some open problems.Comment: 38 pages, 1 figur