The monotone circuit complexity of boolean functions


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(m s/(logm)2s) for fixeds, and sizem Ω(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(n 1/4 · (logn)1/2)), improving a recent result of exp (Ω(n 1/8-ε)) due to Andreev.

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7. References

  1. [1]

    A. V. Aho, J. E. Hopcroft, andJ. D. Ullman,The Design and Analysis of Computer Algorithms, Addison—Wesley, Reading, 1974.

    Google Scholar 

  2. [2]

    A. E. Andreev, On a method for obtaining lower bounds for the complexity of individual monotone functions,Dokl. Ak. Nauk. SSSR 282 (1985), 1033–1037 (in Russian). English translation inSov. Math. Dokl. 31 (1985), 530–534.

    Google Scholar 

  3. [3]

    P. A. Bloniarz,The complexity of monotone Boolean functions and an algorithm for finding shortest paths in a graph, Ph. D. Dissertation, Technical Report238, Laboratory for Computer Science, Massachusetts Institute of Technology, 1979.

  4. [4]

    N. Blum, A Boolean function requiring 3n network size,Theoretical Computer Science 28 (1984), 337–345.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    F. Chung andR. M. Karp,in: Open problems proposed at the NSF Conf. on Complexity Theory, Eugene, Oregon 1984,SIGACT News 16 (1984), 46.

    Google Scholar 

  6. [6]

    P. Erdős andR. Rado, Intersection theorems for systems of sets,Journal of London Mathematical Society 35 (1960), 85–90.

    Article  Google Scholar 

  7. [7]

    J. E. Hopcroft andR. M. Karp, Ann 5/2 algorithm for maximum matching in bipartite graphs,SIAM Journal on Computing 4 (1973), 225–231.

    Article  MathSciNet  Google Scholar 

  8. [8]

    K. Mehlhorn andZ. Galil, Monotone switching circuits and Boolean matrix product,Computing 16 (1976), 99–111.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    J. Nešetřil andS. Poljak, On the complexity of the subgraph problem,CMUC 26 (1985), 415–419.

    Google Scholar 

  10. [10]

    M. S. Paterson, Complexity of monotone networks for Boolean matrix products,Theoretical Computer Science 1 (1975), 13–20.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    V. R. Pratt, The power of negative thinking in multiplying Boolean matrices,SIAM Journal on Computing 4 (1974), 326–330.

    Article  MathSciNet  Google Scholar 

  12. [12]

    A. A. Razborov, Lower bounds for the monotone complexity of some Boolean functions,Dokl. Ak. Nauk. SSSR,281, (1985), 798–801 (in Russian). English translation in:Sov. Math. Dokl.,31 (1985), 354–357.

    MathSciNet  Google Scholar 

  13. [13]

    A. A. Razborov, Lower bounds on monotone network complexity of the logical permanent,Mat. Zametki,37 (1985), 887–900 (in Russian). English translation in:Math. Notes of the Academy of Sciences of the USSR 37 (1985), 485–493.

    MathSciNet  Google Scholar 

  14. [14]

    C. E. Shannon, The synthesis of two-terminal switching circuits,Bell System Technical Journal,28 (1949), 59–98.

    MathSciNet  Google Scholar 

  15. [15]

    S. Skyum andL. G. Valiant, A complexity theory based on Boolean algebra,Journal of the ACM,32:2 (1985), 484–502.

    MATH  Article  MathSciNet  Google Scholar 

  16. [16]

    J. Tiekenheinrich, A 4n-lower bound on the monotone network complexity of a one-output Boolean function,Information Processing Letters,18 (1984), 201–202.

    MATH  Article  MathSciNet  Google Scholar 

  17. [17]

    L. G. Valiant, Completeness classes in algebra,Proceedings of 11 th ACM Symposium on Theory of Computing, (1979), 249–261.

  18. [18]

    I. Wegener, Boolean functions whose monotone complexity is of sizen 2/logn, Theoretical Computer Science,21 (1982), 213–224.

    MATH  Article  MathSciNet  Google Scholar 

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Additional information

First author supported in part by Allon Fellowship, by Bat Sheva de-Rotschild Foundation by the Fund for basic research administered by the Israel Academy of Sciences.

Second author supported in part by a National Science Foundation Graduate Fellowship.

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Alon, N., Boppana, R.B. The monotone circuit complexity of boolean functions. Combinatorica 7, 1–22 (1987).

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AMS subject classification (1980)

  • 68 E 10
  • 68 C 25