Survey propagation an algorithm for satisfiability pdf

Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link. Need an account? Click here to sign up. Download Free PDF. Survey propagation: an algorithm for satisfiability Computing Research Repository, Marc Mezard. Riccardo Zecchina. Alfredo Braunstein. A short summary of this paper. Survey propagation: an algorithm for satisfiability. Braunstein,1,3 M. DOI This algorithm is iterative and composed of two main parts.

Random Struct. However, experimental studies show that Correspondence to: R. Note also the interesting algorithm-independent upper bound found in [1, 28] using the second moment method, which becomes better for larger values of K.

These non-rigorous analytical calculations have put forward some interesting conjectures about what happens in the solution space of the problem as this threshold is approached [22, 24] see also previous work in [2, 25]. Because of this clustering effect, local search algorithms tend to have a very slow convergence when applied to large N instances.

The aim of this paper is to provide a detailed self-contained description of this algorithm, which does not rely on the statistical physics background. The basic building block of the algorithm, called survey propagation SP , is a message passing procedure which resembles in some respect the iterative algorithm known as belief propagation BP , but with some crucial differences which will be described.

While in simple limits we are able to give some rigorous results together with an explicit comparison with the belief propagation procedures, in general there exists no rigorous proof of convergence of the algorithm. However, we provide clear numerical evidence of its performance over benchmarks problems which appear to be far larger than those which can be handled by present state-of-the-art algorithms.

Section 3 explains two message passing algorithms, namely warning propagation WP and belief propagation BP. Both are exact for tree factor graphs.

In Section 5 we give some heuristic arguments from statistical physics which may help the reader to understand where the SP algorithm comes from. Section 6 contains a few general comments. Each constraint is a clause, which is the logical OR of the variables or of their negations. A clause a is characterized by the set of variables i1 ,. In what follows we shall adopt the factor graph representation [18] of the SAT problem. This representation is convenient because it provides an easy graphical description to the message passing procedures which we shall develop.

The SAT problem can be represented graphically as follows see Fig. A function node a is connected to a variable node i by an edge whenever the variable xi or its negation appears in the clause a.

In the graphical representation, we use a full line between a and i whenever the variable appearing in the clause is xi i. Throughout this paper, the variable nodes indices are taken in i, j, k,. A function node a of the factor graph with the Vau j and Vas j sets relative to node j.

Here we shall describe two message passing algorithms. The second algorithm, called belief propagation BP , computes the number of satisfying assignments, as well as the fraction of these assignments where a given variable is set to true. The interpretation of the messages and the message-passing procedure is the following. An example of the use of WP is shown in Fig. The warning propagation algorithm can be applied to any SAT problem.

The interest in WP largely comes from the fact that it gives the exact solution for tree- problems. This is summarized in the following simple theorem: Theorem 1. Consider an instance of the SAT problem with N variables for which the factor graph is a tree.

Run WP 1. Else: 1. GOTO label 1. GOTO label 1 2. Proof of Theorem 1 and of the corollary. The convergence of message passing procedures on tree graphs is a well known result see, e. Call E the set of nodes. If all the ci vanish, the formula is SAT.

Following the step 2. We will also consider the space of SAT assignments with uniform probability for a graph built from the original one by removing a given clause a. Note that this procedure corresponds simply to the application of the right-hand side of Eqs. As WP, the BP algorithm is exact on trees see, for instance, [18]. A working example of BP is shown in Fig. In this example and more in general for trees, BP also provides the exact number N of SAT assignments, as given by the following theorem: Theorem 2.

Comparing to the WP result of Fig. These are the exact results as can be checked by considering all satisfying assignments as in Fig. We just outline the main lines of their proofs for completeness. An alternative proof can be obtained by induction on the level of the edge a, i. The slightly more involved result is the one concerning the entropy. View 1 excerpt. Boolean Model. A new look at survey propagation and its generalizations. SODA ' Threshold values of random K-SAT from the cavity method.

Random Struct. View 1 excerpt, references background. Generalized Belief Propagation. Computer Science, Physics. On the survey-propagation equations for the random K-satisfiability problem.

Local search strategies for satisfiability testing. Cliques, Coloring, and Satisfiability. Factor graphs and the sum-product algorithm. IEEE Trans. Finding hard instances of the satisfiability problem: A survey. Publication Type. More Filters. Empirical investigation of stochastic local search for maximum satisfiability. Frontiers of Computer Science. View 1 excerpt, cites background. View 2 excerpts, cites background. Mathematics, Computer Science.

A continuous-time MaxSAT solver with high analog performance. Computer Science, Medicine. Nature Communications. It is widely acknowledged that stochastic local search SLS algorithms can efficiently find models for satisfiable instances of the satisfiability SAT problem, especially for randomk-SAT … Expand. Planning with preferences using maximum satisfiability. An Improved ID3 classification algorithm for solving the backbone of proposition formulae.

Leveraging cluster backbones for improving MAP inference in statistical relational models. Annals of Mathematics and Artificial Intelligence.