ON THE COMPLEXITY OF RANDOM SATISFIABILITY PROBLEMS WITH PLANTED SOLUTIONS

The problem of identifying a planted assignment given a random k-satisfiability (k-SAT) formula consistent with the assignment exhibits a large algorithmic gap: while the planted solution becomes unique and can 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 sigma 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, for statistical (query) algorithms [M. Kearns, J. ACM, 45 (1998), pp. 983{1006; V. Feldman, E. Grigorescu, L. Reyzin, S. S. Vempala, and Y. Xiao, J. ACM, 64 (2017), pp. 8:1{8:37], matching known upper bounds, which, as we show, can be implemented using a statistical algorithm. Since known approaches for problems over distributions have statistical analogues (spectral, Markov Chain Monte Carlo, gradient-based, convex optimization, etc.), this lower bound provides a rigorous explanation of the observed algorithmic gap. The proof introduces a new general technique for the analysis of statistical query algorithms. It also points to a geometric paring phenomenon in the space of all planted assignments. We describe consequences of our lower bounds to Feige's refutation hypothesis [U. Feige, Proceedings of the ACM Symposium on Theory of Computing, 2002, pp. 534{543] and to lower bounds on general convex programs that solve planted k-SAT. Our bounds also extend to other planted k-CSP models and, in particular, provide concrete evidence for the security of Goldreich's one-way function and the associated pseudorandom generator when used with a sufficiently hard predicate [O. Goldreich, preprint, ia.cr/2000/063, 2000].