See Harry Hsu. “An algorithm for finding a minimal equivalent graph of a digraph.”, Journal of the ACM, 22(1):11-16, January 1975. The simple cubic algorithm below (using an N x N path matrix) suffices for DAGs, but Hsu generalizes it to cyclic graphs.
// reflexive reduction
for (int i = 0; i < N; ++i)
m[i][i] = false;
// transitive reduction
for (int j = 0; j < N; ++j)
for (int i = 0; i < N; ++i)
if (m[i][j])
for (int k = 0; k < N; ++k)
if (m[j][k])
m[i][k] = false;