A study of the binary and Boolean rank of matrices with small constant real rank

We initiate the study of the binary and Boolean rank of 0 , 1 matrices that have a small rank over the reals. The relationship between these three rank functions is an important open question, and here we prove that when the real rank d is a small constant, the gap between the real and the binary and Boolean rank is a small constant. We give tight upper and lower bounds on the Boolean and binary rank of matrices with real rank 1 ≤ d ≤ 4 , as well as determine the size of the largest isolation set in each case. Furthermore, we prove that for d = 3 , 4 , the circulant matrix defined by a row with d − 1 consecutive ones followed by d − 1 zeros, is the only matrix, up to a permutation of the rows and columns and the transpose operation, of size ( 2 d − 2 ) × ( 2 d − 2 ) with real rank d and Boolean and binary rank and isolation set of size 2 d − 2 , and this matrix achieves the maximal gap possible between the real and the binary and Boolean rank for these values of d . Our results can also be interpreted in other equivalent forms, such as finding the minimum number of bicliques needed to partition or cover the edges of a bipartite graph whose reduced adjacency matrix has real rank 1 ≤ d ≤ 4 . We use a combination of combinatorial and algebraic techniques combined with the assistance of a computer program.