Communication lower bounds via the chromatic number

Abstract

We present a new method for obtaining lower bounds on communication complexity. Our method is based on associating with a binary function f a graph Gf such that logχ(Gf) captures N0(f) + N1(f). Here χ(G) denotes the chromatic number of G, and N 0(f) and N1(f) denote, respectively, the nondeterministic communication complexity of f̄ and f. Thus logχ(Gf) is a lower bound on the deterministic as well as zero-error randomized communication complexity of f. Our characterization opens the possibility of using various relaxations of the chromatic number as lower bound techniques for communication complexity. In particular, we show how various (known) lower bounds can be derived by employing the clique number, the Lovász v-function, and graph entropy lower bounds on the chromatic number. © Springer-Verlag Berlin Heidelberg 2007.