The monotone circuit complexity of Boolean functions
Combinatorica
The complexity of Boolean functions
The complexity of Boolean functions
On the method of approximations
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Monotone circuits for connectivity have depth (log n)2-o(1) (extended abstract)
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Counting bottlenecks to show monotone P ? NP
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
A lower bound for the monotone depth of connectivity
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Higher lower bounds on monotone size
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Finite Limits and Monotone Computations: The Lower Bounds Criterion
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
Tight bounds for monotone switching networks via fourier analysis
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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Both Haken's bottleneck counting argument and Razborov's approximation method have been used to prove exponential lower bounds for monotone circuits. We show that under the monotone circuit model for every proof by the approximation method, there is a bottleneck counting proof and vice versa. We also illustrate the elegance of the bottleneck counting technique with a simple self-explained example: the proof of a (previously known) lower bound for the 3-CLIQUE_n problem by the bottleneck counting argument.