On minimum witnesses for boolean matrix multiplication

Minimum witnesses for Boolean matrix multiplication play an important role in several graph algorithms. For two Boolean matrices A and B of order n, with one of the matrices having at most m nonzero entries, the fastest known algorithms for computing the minimum witnesses of their product run in either O(n ^2.575) time or in O(n ²+mnlog(n ²/m)/ log² n) time. We present a new algorithm for this problem. Our algorithm runs either in time Õ(n 3/4-ω m 1-1/4-ω) where ω<2.376 is the matrix multiplication exponent, or, if fast rectangular matrix multiplication is used, in time O (n^1.939m^0.318). In particular, if ω-1<α<2 where m=n ^α, the new algorithm is faster than both of the aforementioned algorithms.