Abstract
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.
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