Some improved bounds on communication complexity via new decomposition of cliques

Abstract

An ordered biclique partition of the complete graph Kn on n vertices is a collection of bicliques (i.e., complete bipartite graphs) such that (i) every edge of Kn is covered by at least one and at most two bicliques in the collection, and (ii) if an edge e is covered by two bicliques then each endpoint of e is in the first class in one of these bicliques and in the second class in the other one. We show in this note that the minimum size of such a collection is O(n2/3).This gives new results on two problems related to communication complexity. Namely, (i) a new separation between the size of a fooling set and the rank of a 0/1-matrix, and (ii) an improved lower bound on the nondeterministic communication complexity of the clique vs. independent set problem are given.