On the Complexity of Random Satisfiability Problems with Planted Solutions (original) (raw)
2015, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15
Random satisfiability problems with planted solutions exhibit an intriguing complexity gap: for problems on n variables with k variables per constraint, after O(n log n) random clauses the planted assignment becomes the unique solution but the best-known algorithms need at least max{n r/2 , n log n} to efficiently identify it (or even one that is correlated with it), for clause distributions that are (r − 1)-wise independent (thus r can be as high as k). We show a nearly tight unconditional lower bound ofΩ(max{n r/2 , n log n}) clauses for any statistical algorithma restricted class of algorithms introduced in [41, 29] that covers most algorithmic approaches commonly used in theory and practice. We complement this with a nearly matching upper bound: a simple, iterative, statistical algorithm that usesÕ(n r/2 ) clauses and time linear in this to find the planted assignment with high probability. As known approaches for planted satisfiability problems (spectral, MCMC, gradient-based, etc.) all have statistical analogues, this provides a rigorous explanation of the large gap between the identifiability and algorithmic identifiability thresholds for random satisfiability problems with planted solutions.
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