Algorithms and certificates for Boolean CSP refutation: smoothed is no harder than random
Venkatesan Guruswami, Pravesh K. Kothari, Peter Manohar
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 n-variable smoothed instance of a k-arity CSP, our algorithm runs in n^O(ℓ) 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 Õ(n) (n/ell)^(k/2 - 1). This matches, up to polylogarithmic factors in n, 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.