Approximating the influence of monotone boolean functions in O(√n) query complexity

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 f : {0, 1}n → {0, 1}, 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 log n/I[f] poly(1/ε)) queries. We also prove a lower bound of Ω(√n/log n·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. © 2011 Springer-Verlag.