SAT and MAX SAT are among the most prominent problems for which local search algorithms have been successfully applied. A fundamental task for such an algorithm is to increase the number of clauses satisfied by a given truth assignment by flipping the truth values of at most k variables (k-fiip local search). For a total number of n variables the size of the search space is of order n~k and grows quickly in k; hence most practical algorithms use 1-flip local search only. In this paper we investigate the worst-case complexity of k-flip local search, considering A; as a parameter: is it possible to search significantly faster than the trivial n~k bound? In addition to the unbounded case we consider instances with a bounded number of literals per clause or where each variable occurs in a bounded number of clauses. We also consider the related problem that asks whether we can satisfy all clauses by flipping the truth values of at most k variables.
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