The asymptotic k-SAT threshold

Abstract

Since the early 2000s physicists have developed an ingenious but non-rigorous formalism called the cavity method to put forward precise conjectures as to the phase transitions in random constraint satisfaction problems ("CSPs"). The cavity method comes in two versions: the simpler replica symmetric variant, and the more intricate 1-step replica symmetry breaking ("1RSB") version. While typically the former only gives upper and lower bounds, the latter is conjectured to yield precise results in many cases. By now, there are a number of examples where the replica symmetric bounds have been verified rigorously. However, verifications of 1RSB predictions are scarce. Perhaps the most prominent challenge in this context is that of pinning down the random k-SAT threshold rk--SAT. Here we prove that rk--SAT = 2k ln 2--1/2 (1 + ln 2) + ok(1), which matches the 1RSB prediction up to the ok(1) error term. The proof directly employs ideas from the 1RSB cavity method, such as the notion of covers (relaxed satisfying assignments) and bits of the Survey Propagation calculations. The best previous lower bound was rk--SAT ≥ 2k ln 2--3/2 ln 2 + ok(1), matching the replica symmetric lower bound asymptotically [Coja-Oghlan, Panagiotou: STOC 2013].