On smoothed k-CNF formulas and the Walksat algorithm
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Smoothed analysis: an attempt to explain the behavior of algorithms in practice
Communications of the ACM - A View of Parallel Computing
Smoothed Analysis of Balancing Networks
ICALP '09 Proceedings of the 36th Internatilonal Collogquium on Automata, Languages and Programming: Part II
Short Propositional Refutations for Dense Random 3CNF Formulas
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
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We introduce the following model for generating semirandom 3CNF formulas. First, an adversary is allowed to pick an arbitrary formula with n variables and m clauses. Then, the formula is slightly perturbed at random. Namely, the smoothing operation leaves the variables of the formula unchanged, but flips the polarity of every variable occurrence in the formula independently with probability \varepsilon . If the density m/n of a 3CNF formula exceeds a certain threshold value (say, 5 \in ^{ - 3} then the smoothing operation almost surely results in a non-satisfiable formula. We present a randomized polynomial time refutation algorithm that for every sufficiently dense 3CNF formula manages to refute most of its smoothed instantiations. The density requirement for our refutation algorithm is roughly\in ^{ - 2} \sqrt {n\log \log n}, which almost matches the density \Omega (\sqrt n ) required by known algorithms for refuting 3CNF formulas that are completely random.