Solving a Special Type of Optimal Transport Problem by a Modified Hungarian Algorithm

We observe that computing empirical Wasserstein distance in the independence test is an optimal transport (OT) problem with a special structure. This observation inspires us to study a special type of OT problem and propose a modified Hungarian algorithm to solve it exactly. For an OT problem between marginals with $m$ and $n$ atoms ($m\geq n$), the computational complexity of the proposed algorithm is $\mathcal{O}(m^2n)$. Computing the empirical Wasserstein distance in the independence test requires solving this special type of OT problem, where we have $m=n^2$. The associate computational complexity of our algorithm is $\mathcal{O}(n^5)$, while the order of applying the classic Hungarian algorithm is $\mathcal{O}(n^6)$. Numerical experiments validate our theoretical analysis. Broader applications of the proposed algorithm are discussed at the end.