Random Formulas Have Frozen Variables

Abstract

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.

The asymptotic k-SAT threshold

“…That is, one can walk within the cluster Ci from any σ ∈ Ci to any other τ ∈ Ci by only altering, say, ln n variables at each step. The existence of clusters and frozen variables has by now been established rigorously [1,4,33].…”

“…Provably, Σ must remain exponentially large w.h.p. right up to rk−SAT [4]. Hence, as clusters are well-separated, there might be a chance that two random clusters decorrelate, even though two randomly chosen satisfying assignments do not.…”

“…4 Yet matters are far from straightforward as the asymmetry of the k-SAT problem implies that, much like satisfying assignments, covers lean towards σmaj and thus are subtly correlated. In effect, a "vanilla" second moment argument cannot succeed.…”

The cook-book approach to the differential equation method

“…In particular, Mezard, Parisi and Zecchina [62,63] stated that the threshold for 3-SAT occurs at clause density c 3 = 4.27. Recently, Achlioptas and Ricci-Tersenghi [5] proved that for k ≥ 8, the thresholds for K-SAT found by use of statistical physics techniques could be considered rigorously proved. In the following we focus on methods giving increasingly improved rigorous lower and upper bounds on that threshold point.…”

VARSAT: Integrating Novel Probabilistic Inference Techniques with DPLL Search

“…To date they have all employed a "decimation" framework of computing a single sequence of estimates, while setting a block of one or more most strongly-biased variables after each estimate. Taken with this decimation framework, the probabilistic bias estimators are state-of-theart for solving large random problems near the critically-constrained phase transition in problem hardness [4]. However, this construction cannot backtrack or take advantage of modern advances in systematic DPLL search like clause learning; the decimation process either directly reaches a solution by a series of fortuitous variable assignments, or it ends in failure without determining satisfiability or unsatisfiability.…”