The monotone circuit complexity of boolean functions

@article{Alon1987TheMC,
  title={The monotone circuit complexity of boolean functions},
  author={N. Alon and R. Boppana},
  journal={Combinatorica},
  year={1987},
  volume={7},
  pages={1-22}
}
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… Expand

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References

SHOWING 1-10 OF 21 REFERENCES
THE COMPLEXITY OF MONOTONE BOOLEAN FUNCTIONS AND AN ALGORITHM FOR FINDING SHORTEST PATHS ON A GRAPH
The first part of this thesis considers the complexity of Boolean functions. The main complexity measures used are the number of gates in combinational networks and the size of Boolean formulas. TheExpand
Boolean Functions Whose Monotone Complexity is of Size n2/log n
  • I. Wegener
  • Mathematics, Computer Science
  • Theoretical Computer Science
  • 1981
TLDR
A sequence of monotone Boolean functions hn:{0, 1}n→{0,1}n, such that the monotones complexity of hn is of order n2/log n is constructed, which includes the largest known lower bound of this kind. Expand
A 4n-Lower Bound on the Monotone Network Complexity of a One-Output Boolean Function
TLDR
In this paper a special function is presented for which a lower bound of size 4n over the monotone basis can be proved. Expand
The power of negative thinking in multiplying Boolean matrices
  • V. Pratt
  • Mathematics, Computer Science
  • STOC '74
  • 1974
TLDR
It is shown that n-input inputs are needed to form the product of two Boolean matrices, and hence O(n-supscrpt) two-input <underline>and-gate inputs are required to compute the transitive closure of a Boolean matrix. Expand
Completeness classes in algebra
TLDR
The aim of this paper is to demonstrate that for both algebraic and combinatorial problems this phenomenon exists in a form that is purely algebraic in both of the respects (A) and (B). Expand
A Boolean Function Requiring 3n Network Size
  • Norbert Blum
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1984
TLDR
Paul (1977) has proved a 2.5 n -lower bound for the network complexity of an explicit Boolean function, but this work modifications the definition of Paul's function slightly and proves a 3 n - lower bound for that function. Expand
A complexity theory based on Boolean algebra
  • Sven Skyum, L. Valiant
  • Mathematics, Computer Science
  • 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981)
  • 1981
TLDR
It is shown that much of what is of everyday relevance in Turing-machine-based complexity theory can be replicated easily and naturally in this elementary framework. Expand
Complexity of Monotone Networks for Boolean Matrix Product
  • M. Paterson
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1975
Any computation of Boolean matrix product by an acyclic network using only the operations of binary conjunction and disjunction requires at least IJK conjunctions and IJ(K-1) disjunctions for the Expand
An n5/2 Algorithm for Maximum Matchings in Bipartite Graphs
The present paper shows how to construct a maximum matching in a bipartite graph with n vertices and m edges in a number of computation steps proportional to $(m + n)\sqrt n $.
The Design and Analysis of Computer Algorithms
TLDR
This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs. Expand
...
1
2
3
...