View article

The problem of identifying a planted assignment given a random k-SAT formula consistent with the assignment exhibits a large algorithmic gap: while the planted solution can always be identified given a formula with O(n log n) clauses, there are distributions over clauses for which the best known efficient algorithms require n^k/2 clauses. We propose and study a unified model for planted k-SAT, which captures well-known special cases. An instance is described by a planted assignment σ and a distribution on clauses with k literals. We define its distribution complexity as the largest r for which the distribution is not r-wise independent (1 ≤ r ≤ k for any distribution with a planted assignment).

Our main result is an unconditional lower bound, tight up to logarithmic factors, of Ω(n^r/2) clauses for statistical algorithms, matching the known upper bound (which, as we show, can be implemented using a statistical algorithm …