An axiomatization of the Banzhaf value
International Journal of Game Theory
Learning monotone ku DNF formulas on product distributions
COLT '91 Proceedings of the fourth annual workshop on Computational learning theory
Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
On the Fourier spectrum of monotone functions
Journal of the ACM (JACM)
The importance of being biased
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
A Fourier-theoretic perspective on the Condorcet paradox and Arrow's theorem
Advances in Applied Mathematics
A Sharp Threshold for Network Reliability
Combinatorics, Probability and Computing
Journal of Computer and System Sciences - Special issue on FOCS 2002
Learning Monotone Decision Trees in Polynomial Time
SIAM Journal on Computing
Collective coin flipping, robust voting schemes and minima of Banzhaf values
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Improved Bounds for Testing Juntas
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Testing juntas nearly optimally
Proceedings of the forty-first annual ACM symposium on Theory of computing
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Testing Computability by Width Two OBDDs
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Bounding the average sensitivity and noise sensitivity of polynomial threshold functions
Proceedings of the forty-second ACM symposium on Theory of computing
A o(n) monotonicity tester for boolean functions over the hypercube
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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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 nċI[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.