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

<i>The Total Influence</i> (<i>Average Sensitivity</i>) 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>I</i>[<i>f</i>]. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of (1 ± <i>ε</i>) by performing <i>O</i>(√<i>n</i> <i>I</i>[<i>f</i>] poly(1/<i>ε</i>)) queries. We also prove a lower bound of <i>Ω</i>(√<i>n</i> log<i>n</i>·<i>I</i>[<i>f</i>]) on the query complexity of any constant factor approximation algorithm for this problem (which holds for <i>I</i>[<i>f</i>]=<i>Ω</i>(1)), hence showing that our algorithm is almost optimal in terms of its dependence on <i>n</i>. For general functions, we give a lower bound of <i>Ω</i>(<i>n</i> <i>I</i>[<i>f</i>]), which matches the complexity of a simple sampling algorithm.

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