This paper studies approximation of the fixed support Wasserstein barycenter problem. The question is how to find the barycenter of a set of discrete probability distributions. While optimal transport can be cast as a min-cost flow, the authors solve an open problem in showing that fixed support Wasserstein barycenter is not. The paper also proposes a fast approximation method and a few experimental results showing a tradeoff between speed and accuracy. Reviewers felt the experimental evaluation could be significantly better but overall this is a great contribution, suitable for Neurips.