{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T11:02:47Z","timestamp":1790506967130,"version":"4.1.0"},"reference-count":85,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1217793"],"award-info":[{"award-number":["CCF-1217793"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>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 ($1 \\le r \\le k$ 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].<\/jats:p>","DOI":"10.1137\/16m1078471","type":"journal-article","created":{"date-parts":[[2018,7,13]],"date-time":"2018-07-13T08:41:38Z","timestamp":1531471298000},"page":"1294-1338","source":"Crossref","is-referenced-by-count":17,"title":["On the Complexity of Random Satisfiability Problems with Planted Solutions"],"prefix":"10.1137","volume":"47","author":[{"given":"Vitaly","family":"Feldman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Will","family":"Perkins","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Santosh","family":"Vempala","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,7,12]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1109\/TNSE.2014.2368716"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.4086\/toc.2015.v011a017"},{"key":"atypb3","first-page":"793","author":"Achlioptas D.","year":"2008","journal-title":"Proceedings of the IEEE Symposium on Foundations of Computer Science"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1613\/jair.1681"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-011-0029-x"},{"key":"atypb6","first-page":"689","author":"Allen S. 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