Closing the Gap: Efficient Algorithms for Discrete Wasserstein Barycenters
Abstract
The Wasserstein barycenter problem seeks a probability measure that minimizes the weighted average of the Wasserstein distances to a given collection of probability measures. We study the discrete setting, where each measure has finite support-- a regime that frequently arises in machine learning and operations research. The discrete Wasserstein barycenter problem is known to be NP-hard, which motivates us to study approximation algorithms with provable guarantees. The best-known algorithm to date achieves an approximation ratio of two. We close this gap by developing a polynomial-time approximation scheme (PTAS) for the discrete Wasserstein barycenter problem that generalizes and improves upon the 2-approximation method. In addition, for the special case of equally weighted measures, we obtain a strictly tighter approximation guarantee. Numerical experiments show that the proposed algorithms are computationally efficient and produce near-optimal barycenter solutions.
Cite
@article{arxiv.2511.04607,
title = {Closing the Gap: Efficient Algorithms for Discrete Wasserstein Barycenters},
author = {Jiaqi Wang and Weijun Xie},
journal= {arXiv preprint arXiv:2511.04607},
year = {2025}
}