Random Formulas Have Frozen Variables

For a large number of random constraint satisfaction problems, such as random k-SAT and random graph and hypergraph coloring, we have very good estimates of the largest constraint density for which solutions exist. Yet, all known polynomial-time algorithms for these problems fail to find solutions even at much lower densities. To understand the origin of this gap one can study how the structure of the space of solutions evolves in such problems as constraints are added. In particular, it is known that much before solutions disappear, they organize into an exponential number of clusters, each of which is relatively small and far apart from all other clusters. Here we further prove that inside every cluster a majority of variables are frozen, i.e., take only one value. The existence of such frozen variables gives a satisfying intuitive explanation for the failure of the polynomial-time algorithms analyzed so far. At the same time, our results lend support to one of the two main hypotheses underlying Survey Propagation, a heuristic introduced by physicists in recent years that appears to perform extraordinarily well on random constraint satisfaction problems.