Abstract
The Total Influence (Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function, which we denote by I[f]. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of (1 ± ε) by performing O(√n I[f] poly(1/ε)) queries. We also prove a lower bound of Ω(√n logn·I[f]) on the query complexity of any constant factor approximation algorithm for this problem (which holds for I[f]=Ω(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 Ω(n I[f]), which matches the complexity of a simple sampling algorithm.
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Approximating the Influence of Monotone Boolean Functions in O(√n) Query Complexity
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