Monotone multilinear boolean circuits for bipartite perfect matching require exponential size

Abstract

A monotone boolean circuit is said to be multilinear if for any AND gate in the circuit, the minimal representation of the two input functions to the gate do not have any variable in common. We show that multilinear boolean circuits for bipartite perfect matching require exponential size. In fact we prove a stronger result by characterizing the structure of the smallest multilinear boolean circuits for the problem. We also show that the upper bound on the minimum depth of monotone circuits for perfect matching in general graphs is O(n). © Springer-Verlag Berlin Heidelberg 2004.