The random K-satisfiability problem: from an analytic solution to an efficient algorithm

Marc Mézard 1, Riccardo Zecchina 2

Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 66 (2002) 056126

We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search algorithms. The fundamental order parameter introduced in the cavity method, which consists of surveys of local magnetic fields in the various possible states of the system, can be computed for one given sample. These surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems. One such algorithm is shown here for the 3-sat problem, with very good performances.

  • 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 2. The Abdus Salam International Centre for Theoretical Physics, Statistical Mechanics and Interdisciplinary Applications Group,
    the Abdus Salam International Centre for Theoretical Physics