On the diameter of the set of satisfying assignments in random satisfiable k-CNF formulas

TL;DR

This paper investigates how the solution space diameter of satisfiable random k-CNF formulas sharply decreases from about n/2 to nearly zero as the clause-variable ratio crosses the satisfiability threshold, revealing a phase transition in solution structure.

Contribution

It establishes that just above the satisfiability threshold, the diameter of the solution space becomes very small, specifically O(k2^{-k}n), highlighting a sudden structural change.

Findings

Diameter drops from ~n/2 to O(k2^{-k}n] above the threshold

Small change in density causes a dramatic reduction in solution space size

Excludes unsatisfiable formulas, focusing on satisfiable instances

Abstract

It is known that random k-CNF formulas have a so-called satisfiability threshold at a density (namely, clause-variable ratio) of roughly 2^k\ln 2: at densities slightly below this threshold almost all k-CNF formulas are satisfiable whereas slightly above this threshold almost no k-CNF formula is satisfiable. In the current work we consider satisfiable random formulas, and inspect another parameter -- the diameter of the solution space (that is the maximal Hamming distance between a pair of satisfying assignments). It was previously shown that for all densities up to a density slightly below the satisfiability threshold the diameter is almost surely at least roughly n/2 (and n at much lower densities). At densities very much higher than the satisfiability threshold, the diameter is almost surely zero (a very dense satisfiable formula is expected to have only one satisfying assignment). In this paper we show that for all densities above a density that is slightly above the satisfiability threshold (more precisely at ratio (1+ \eps)2^k \ln 2, \eps=\eps(k) tending to 0 as k grows) the diameter is almost surely O(k2^{-k}n). This shows that a relatively small change in the density around the satisfiability threshold (a multiplicative (1 + \eps) factor), makes a dramatic change in the diameter. This drop in the diameter cannot be attributed to the fact that a larger fraction of the formulas is not satisfiable (and hence have diameter 0), because the non-satisfiable formulas are excluded from consideration by our conditioning that the formula is satisfiable.