Optimal circuits and transitive automorphism groups
In this paper we ask, “Can anything be said about a smallest circuit that computes a given function.” We are able to show that for a wide class of functions, which includes all graph problems, an optimal circuit has a restricted structure. For instance, the input wires in an optimal circuit for Hamiltonian Path have at most linear fan-out. This is analogous to the possibly counter-intuitive statement that there is a straight line program for Hamiltonian Path on graphs with n nodes that looks at each edge only O(n2) times.
KeywordsBoolean Function Hamiltonian Path Graph Function Input String Graph Problem
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- [Boppana]R. B. Boppana, Threshold Functions and Rounded Depth Monotone Circuits, ACM SIGACT 16(1984), 475–479.Google Scholar
- [Fagin]R. Fagin, M. M. Klawe, N. J. Pippenger, L. Stockmeyer. Bounded depth. polynomial-size circuits for symmetric functions, IBM Research Report RI 4040. Oct. 1983.Google Scholar
- [Furst]M. Furst, J. B. Saxe, and M. Sipser, Parity, Circuits, and the Polynomial Time Hierarchy, IEEE FOCS 22(1981) 260–270.Google Scholar
- [Paul]W. J. Paul, Realizing Boolean Functions on Disjoint Sets of Variables. Theoretical Computer Science 2(1976) 383–396.Google Scholar
- [Smolensky]R. Smolensky, Algebraic Methods in the Theory of Lower Bounds for Boolean Circuit Complexity, ACM SIGACT 19(1987).Google Scholar