Approximating the influence of monotone Boolean functions in O(√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 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 ± ∈) by performing O (equation) queries. We also prove a lower bound of Ω (equation) 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.