Abstract

It is proved here that every monotone circuit which tests $st$-connectivity of an undirected graph on n nodes has depth $\Omega (\log^2 \,n)$. This implies a superpolynomial $(n^{\Omega (\log n)} )$ lower bound on the size of any monotone formula for $st$-connectivity.
The proof draws intuition from a new characterization of circuit depth in terms of communication complexity. Within the same framework, a very simple and intuitive proof is given of a depth analogue of a theorem of Khrapchenko concerning formula size lower bounds.

MSC codes

  1. 06E30
  2. 94A05
  3. 94C10

Keywords

  1. circuit complexity
  2. communication complexity
  3. monotone circuits
  4. graph connectivity

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Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 255 - 265
ISSN (online): 1095-7146

History

Submitted: 1 March 1989
Accepted: 9 June 1989
Published online: 8 August 2006

MSC codes

  1. 06E30
  2. 94A05
  3. 94C10

Keywords

  1. circuit complexity
  2. communication complexity
  3. monotone circuits
  4. graph connectivity

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