On the survey-propagation equations in random constraint satisfiability problems

In this paper we study the existence of a solution of the survey-propagation equations for a given instance of a random problem in the framework of constraint satisfiability. We consider the concrete examples of $K$-satisfiability and coloring. We conjecture that when the number of variables goes to infinity, the solution of the survey-propagation equations for a given instance can be obtained by finding the (supposed unique) solution of the corresponding equations on an infinite tree. We conjecture that the survey-propagation equations on a random infinite tree have a unique solution in the suitable range of parameters. We also present analytic arguments that indicate that the survey-propagation equations do have solutions in the satisfiable phase. For simplicity of notation the argument is presented in the case of the coloring problem. The same results extend to other optimization problems where exist configurations that have cost zero, i.e., in the satisfiable phase. On a random graph the solutions of the belief-propagation equations are associated with the existence of many well separated clusters of configurations (clustering states). We argue that on a random graph the belief-propagation equations have solutions almost everywhere: the statement may be sharpened by introducing the concept of quasisolution of the belief-propagation equations.