2019/05/28 by Nathaniel Lahn, Lahn, Nathaniel, Deepika Mulchandani +3 · 2 citations
Computer Science · #Complexity and Algorithms in Graphs #Data Management and Algorithms #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Optimization and Search Problems
paper · pdf · doi:10.48550/arxiv.1905.11830
openalex publication_date 2019/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 near-linear time multiplicative approximation algorithms. There has also been an effort to design approximate solutions with additive errors [Cuturi NIPS '13, Altschuler \etal NIPS '17, Dvurechensky \etal ICML '18, Quanrud, SOSA '19] within a time bound that is linear in the size of the cost matrix and polynomial in C/δ; here C is the largest value in the cost matrix and δ 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(\fracn2 Cδ+ (nC2)/(δ2)). Our algorithm is extremely simple and executes, for an arbitrarily small constant ε, only \lfloor (2C)/((1-ε)δ)\rfloor + 1 iterations, where each iteration consists only of a Dijkstra-type search followed by a depth-first search. We also provide empirical results that suggest our algorithm is competitive with respect to a sequential implementation of the Sinkhorn algorithm in execution time. Moreover, our algorithm quickly computes a solution for very small values of δ whereas Sinkhorn algorithm slows down due to numerical instability.