The complexity of Boolean functions
The complexity of Boolean functions
The complexity of Boolean networks
The complexity of Boolean networks
Poceedings of the London Mathematical Society symposium on Boolean function complexity
On lower bounds for read-k-times branching programs
Computational Complexity
Lower bounds on arithmetic circuits via partial derivatives
Computational Complexity
Branching programs and binary decision diagrams: theory and applications
Branching programs and binary decision diagrams: theory and applications
Models of Computation: Exploring the Power of Computing
Models of Computation: Exploring the Power of Computing
Introduction to Coding Theory
A characterization of span program size and improved lower bounds for monotone span programs
Computational Complexity
A note on monotone complexity and the rank of matrices
Information Processing Letters
Multi-linear formulas for permanent and determinant are of super-polynomial size
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Multilinear- NC" " Multilinear- NC"
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Monotone multilinear boolean circuits for bipartite perfect matching require exponential size
FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
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We prove optimal lower bounds for multilinear circuits and for monotone circuits with bounded depth. These lower bounds state that, in order to compute certain functions, these circuits need exactly as many OR gates as the respective DNFs. The proofs exploit a property of the functions that is based solely on prime implicant structure. Due to this feature, the lower bounds proved also hold for approximations of the considered functions that are similar to slice functions. Known lower bound arguments cannot handle these kinds of approximations. In order to show limitations of our approach, we prove that cliques of size n–1 can be detected in a graph with n vertices by monotone formulae with O(log n) OR gates. Our lower bound for multilinear circuits improves a lower bound due to Borodin, Razborov and Smolensky for nondeterministic read-once branching programs computing the clique function.