From one solution of a 3-satisfiability formula to a solution cluster: frozen variables and entropy.

A solution to a 3-satisfiability (3-SAT) formula can be expanded into a cluster, all other solutions of which are reachable from this one through a sequence of single-spin flips. Some variables in the solution cluster are frozen to the same spin values by one of two different mechanisms: frozen-core formation and long-range frustrations. While frozen cores are identified by a local whitening algorithm, long-range frustrations are very difficult to trace, and they make an entropic belief-propagation (BP) algorithm fail to converge. For the BP algorithm to reach a fixed point, the spin values of a tiny fraction of variables (chosen according to the whitening algorithm) are externally fixed during the iteration. From the calculated entropy values, we infer that, for a large random 3-SAT formula with constraint density close to the satisfiability threshold, the solutions obtained by the survey-propagation or WALKSAT algorithms belong neither to the most dominating clusters of the formula nor to the most abundant clusters. This work indicates that a single-solution cluster of a random 3-SAT formula may have further community structures.