Witnesses for Boolean Matrix Multiplication and for Transitive Closure

The subcubic (O(nω) for ω < 3) algorithms to multiply Boolean matrices do not provide the witnesses; namely, they compute C = A · B but if Cij = 1 they do not find an index k (a witness) such that Aik = Bkj = 1. We design a deterministic algorithm for computing the matrix of witnesses which runs in O(nω + ϵ) time for any positive e. We also design an algorithm that computes witnesses for the transitive closure in the same time needed to compute witnesses for Boolean matrix multiplication.