Clustering of solutions in the random satisfiability problem

M. Mezard ¹, T. Mora ¹, R. Zecchina ²

Physical Review Letters 94 (2005) 197205

Using elementary rigorous methods we prove the existence of a clustered phase in the random $K$-SAT problem, for $K\\geq 8$. In this phase the solutions are grouped into clusters which are far away from each other. The results are in agreement with previous predictions of the cavity method and give a rigorous confirmation to one of its main building blocks. It can be generalized to other systems of both physical and computational interest.

• 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,
  ICTP Trieste