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 \(f: \{0,1\}^n \longrightarrow \{0,1\}\), 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 \(O\left(\frac{\sqrt{n}\log n}{I[f]}{\rm poly}(1/\epsilon) \right) \) queries. We also prove a lower bound of \(\Omega\left(\frac{\sqrt{n}}{\log n \cdot I[f]}\right)\) 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 \(\Omega\left(\frac{n}{I[f]}\right)\), which matches the complexity of a simple sampling algorithm.
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Ron, D., Rubinfeld, R., Safra, M., Weinstein, O. (2011). Approximating the Influence of Monotone Boolean Functions in \(O(\sqrt{n})\) Query Complexity. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2011 2011. Lecture Notes in Computer Science, vol 6845. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22935-0_56
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