Approximating the Influence of Monotone Boolean Functions in $O(\sqrt{n})$ Query Complexity

The Total Influence (Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influe nce of a monotone Boolean function f : f0;1g n ! f 0;1g, which we denote by I[f]. We present a randomized algorithm that approximates the influence of such functions to within a multip licative factor of (1 � �) by performing O � p nlog n I[f] poly(1=�) � queries. We also prove a lower bound of � p n log n·I[f] � on the query complexity of any constant-factor approximation algorithm for this problem (which holds for I[f] = (1)), hence showing that our algorithm is almost optimal in terms of its dependence on n. For general functions we give a lower bound of � n I[f] � , which matches the complexity of a simple sampling algorithm.

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