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Multi-marginal optimal transport (MOT) is a generalization of optimaltransport to multiple marginals. Optimal transport has evolved into animportant tool in many machine learning applications, and its multi-marginalextension opens up for addressing new challenges in the field of machinelearning. However, the usage of MOT has been largely impeded by itscomputational complexity which scales exponentially in the number of marginals.Fortunately, in many applications, such as barycenter or interpolationproblems, the cost function adheres to structures, which has recently beenexploited for developing efficient computational methods. In this work wederive computational bounds for these methods. With $m$ marginal distributionssupported on $n$ points, we provide a $ \mathcal{\tilde O}(d(G)mn^2\epsilon^{-2})$ bound for a $\epsilon$-accuracy when the problem isassociated with a tree with diameter $d(G)$. For the special case of theWasserstein barycenter problem, which corresponds to a star-shaped tree, ourbound is in alignment with the existing complexity bound for it.

On the complexity of the optimal transport problem with graph-structured cost