In the 1960s, it was recognized that the discontinuity structure of Feynman integrals is highly constrained by a condition known as the hierarchical principle, which specifies which pairs of double discontinuities must vanish for a given Feynman integral. Despite its significance, developing a practical algorithm for working out these constraints has proven challenging. Building on recent advances in the study of singularities of Feynman integrals, I will present an efficient algorithm for applying the hierarchical principle. I will illustrate the method with simple one-loop examples before demonstrating its capability to algorithmically derive $\mathcal{O}(100)$ constraints in computation times on the order of minutes for two-loop Feynman integrals recently evaluated for processes critical to precision predictions for collider physics.