, Volume 7, Issue 1, pp 1–22

The monotone circuit complexity of boolean functions

  • Noga Alon
  • Ravi B. Boppana

DOI: 10.1007/BF02579196

Cite this article as:
Alon, N. & Boppana, R.B. Combinatorica (1987) 7: 1. doi:10.1007/BF02579196


Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that cliques in graphs. In particular, Razborov showed that detecting cliques of sizes in a graphm vertices requires monotone circuits of size Ω(ms/(logm)2s) for fixeds, and sizemΩ(logm) form/4].

In this paper we modify the arguments of Razborov to obtain exponential lower bounds for circuits. In particular, detecting cliques of size (1/4) (m/logm)2/3 requires monotone circuits exp (Ω((m/logm)1/3)). For fixeds, any monotone circuit that detects cliques of sizes requiresm)s) AND gates. We show that even a very rough approximation of the maximum clique of a graph requires superpolynomial size monotone circuits, and give lower bounds for some Boolean functions. Our best lower bound for an NP function ofn variables is exp (Ω(n1/4 · (logn)1/2)), improving a recent result of exp (Ω(n1/8-ε)) due to Andreev.

AMS subject classification (1980)

68 E 1068 C 25

Copyright information

© Akadémiai Kiadó 1987

Authors and Affiliations

  • Noga Alon
    • 1
    • 2
  • Ravi B. Boppana
    • 3
  1. 1.Department of MathematicsTel Aviv UniversityTel AvivIsrael
  2. 2.IBM Almaden Research CenterSan JoseUSA
  3. 3.Laboratory for Computer ScienceMassachusetts Inst. of Tech.CambridgeUSA