Abstract
We prove tight size bounds on monotone switching networks for the NP-complete problem of $k$-clique, and for an explicit monotone problem by analyzing a pyramid structure of height $h$ for the P-complete problem of generation. This gives alternative proofs of the separations of m-NC from m-P and of m-NC$^i$ from m-NC$^{i+1}$, different from Raz--McKenzie (Combinatorica 1999). The enumerative-combinatorial and Fourier analytic techniques in this paper are very different from a large body of work on circuit depth lower bounds, and may be of independent interest. An earlier version of this paper appeared in the Proceedings of the 44th ACM Symp. on Theory of Computing, pp. 495-504, 2011.
Highlights
To study parallel time and memory usage complexity, lower bounds are sought in different models of computation
As for the parallel time of efficiently computable functions, Karchmer and Wigderson [34] showed that the directed connectivity problem requires super-logarithmic (Ω(log2 n)) depth for monotone circuits,1 implying m-NC1 m-non-deterministic logspace (NL) ⊆ m-NC2
(1) the complete problem for NL, directed connectivity, requires Ω(log2 n) depth, reproving the tight bound of Karchmer and Wigderson [34]; (2) the “complete problem for NCi,” the Generation problem with a pyramid structure of height h, requires Ω(h log n) depth when h ≤ nc for some constant c > 0, giving m-Nick’s class (NC) = m-P and m-NCi = m-NCi+1 for all i by setting h = logi n; and
Summary
To study parallel time and memory usage complexity, lower bounds are sought in different models of computation. As for the parallel time of efficiently computable functions, Karchmer and Wigderson [34] showed that the directed connectivity problem requires super-logarithmic (Ω(log n)) depth for monotone circuits, implying m-NC1 m-NL ⊆ m-NC2. Raz and McKenzie [46] extended the lower bound framework of Karchmer and Wigderson [34], proving that a monotone circuit computing (1) the complete problem for NL, directed connectivity, requires Ω(log n) depth, reproving the tight bound of Karchmer and Wigderson [34];. From the best lower bounds for the depth of monotone circuits by Raz and McKenzie [46], it follows that a monotone switching network computing (1’) directed connectivity requires nΩ(1) size; √. Potechin [45] introduced a Fourier analytic framework for analyzing extremal instances (minterms and maxterms) of the monotone Boolean function of directed connectivity
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