Algorithms and Certificates for Boolean CSP Refutation: Smoothed Is No Harder than Random

Abstract

SIAM Journal on Computing, Ahead of Print.
Abstract. We present an algorithm for strongly refuting smoothed instances of all Boolean CSPs. The smoothed model is a hybrid between worst- and average-case input models, where the input is an arbitrary instance of the CSP with only the negation patterns of the literals re-randomized with some small probability. For an [math]-variable smoothed instance of a [math]-arity CSP, our algorithm runs in [math] time and succeeds with high probability in bounding the optimum fraction of satisfiable constraints away from 1, provided that the number of constraints is at least [math]. This matches, up to polylogarithmic factors in [math], the trade-off between running time and the number of constraints of the state-of-the-art algorithms for refuting fully random instances of CSPs [P. Raghavendra, S. Rao, and T. Schramm, Strongly refuting random CSPs below the spectral threshold, in STOC’17—Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, 2017, pp. 121–131]. We also make a surprising connection between the analysis of our refutation algorithm in the significantly “randomness starved” setting of semirandom [math]-XOR and the existence of even covers in worst-case hypergraphs. We use this connection to positively resolve Feige’s 2008 conjecture—an extremal combinatorics conjecture on the existence of even covers in sufficiently dense hypergraphs that generalizes the well-known Moore bound for the girth of graphs. As a corollary, we show that polynomial-size refutation witnesses exist for arbitrary smoothed CSP instances with number of constraints a polynomial factor below the “spectral threshold” of [math], extending the celebrated result for random 3-SAT of [U. Feige, J. H. Kim, and E. Ofek, Witnesses for non-satisfiability of dense random 3CNF formulas, in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, 2006, pp. 497–508].