The Rectangle Covering Number of Random Boolean Matrices

The rectangle covering number of an n x n Boolean matrix M is the smallest number of 1-rectangles which are needed to cover all the 1-entries of M. Its binary logarithm is the Nondeterministic Communication Complexity, and it equals the chromatic number of a graph G boxed times(M) obtained from M by a construction of Lovasz and Saks. We determine the rectangle covering number and related parameters (clique size, independence ratio, fractional chromatic number of G boxed times(M)) of random Boolean matrices, where each entry is 1 with probability p = p (n), and the entries are independent.