A Graph Theoretic Additive Approximation of Optimal Transport

Transportation cost is an attractive similarity measure between probability distributions due to its many useful theoretical properties. However, solving optimal transport exactly can be prohibitively expensive. Therefore, there has been significant effort towards the design of scalable approximation algorithms. Previous combinatorial results [Sharathkumar, Agarwal STOC '12, Agarwal, Sharathkumar STOC '14] have focused primarily on the design of strongly polynomial multiplicative approximation algorithms. There has also been an effort to design approximate solutions with additive errors [Cuturi NIPS '13, Altschuler et. al NIPS '17, Dvurechensky et al., ICML '18, Quanrud, SOSA '19] within a time bound that is linear in the size of the cost matrix and polynomial in $C/\delta$; here $C$ is the largest value in the cost matrix and $\delta$ is the additive error. We present an adaptation of the classical graph algorithm of Gabow and Tarjan and provide a novel analysis of this algorithm that bounds its execution time by $O(\frac{n^2 C}{\delta}+ \frac{nC^2}{\delta^2})$. Our algorithm is extremely simple and executes, for an arbitrarily small constant $\varepsilon$, only $\lfloor \frac{2C}{(1-\varepsilon)\delta}\rfloor + 1$ iterations, where each iteration consists only of a Dijkstra search followed by a depth-first search. We also provide empirical results that suggest our algorithm significantly outperforms existing approaches in execution time.