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Survey Propagation is an algorithm designed for solving typical instances of random constraint satisfiability problems. It has been successfully tested on random 3-SAT and random $G(n,\frac{c}{n})$ graph 3-coloring, in the hard region of the parameter space. Here we provide a generic formalism which applies to a wide class of discrete Constraint Satisfaction Problems., Comment: 8 pages, 5 figures

We study the graph coloring problem over random graphs of finite average connectivity $c$. Given a number $q$ of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on $q$, we find the precise value of the critical average connectivity $c_q$. Moreover, we show that below $c_q$ there exist a clustering phase $c\in [c_d,c_q]$ in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms., Comment: 4 pages, 1 figure, version to app. in PRL

A major problem in evaluating stochastic local search algorithms for NP-complete problems is the need for a systematic generation of hard test instances having previously known properties of the optimal solutions. On the basis of statistical mechanics results, we propose random generators of hard and satisfiable instances for the 3-satisfiability problem (3SAT). The design of the hardest problem instances is based on the existence of a first order ferromagnetic phase transition and the glassy nature of excited states. The analytical predictions are corroborated by numerical results obtained from complete as well as stochastic local algorithms., Comment: 5 pages, 4 figures, revised version to app. in PRL

We study a simple and exactly solvable model for the generation of random satisfiability problems. These consist of $\gamma N$ random boolean constraints which are to be satisfied simultaneously by $N$ logical variables. In statistical-mechanics language, the considered model can be seen as a diluted p-spin model at zero temperature. While such problems become extraordinarily hard to solve by local search methods in a large region of the parameter space, still at least one solution may be superimposed by construction. The statistical properties of the model can be studied exactly by the replica method and each single instance can be analyzed in polynomial time by a simple global solution method. The geometrical/topological structures responsible for dynamic and static phase transitions as well as for the onset of computational complexity in local search method are thoroughly analyzed. Numerical analysis on very large samples allows for a precise characterization of the critical scaling behaviour., Comment: 14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor errors and references corrected