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In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576]. 

Abstract Assessing the true lineage diversity in elusive nocturnal organisms is particularly challenging due to their subtle phenotypic variation in diagnostic traits. The cryptic small-eared greater galago (Otolemur garnettii) offers a great opportunity to test if currently recognized subspecies, suggested by discontinuities in coat colour pattern and geographic barriers, represent distinct evolutionary lineages. To answer this question, we conducted the first population-level phylogeographic study of the species, sampling wild specimens from across almost its entire latitudinal range, including the Zanzibar Archipelago. We applied five species-delimitation algorithms to investigate the genetic diversity and distribution pattern of mitochondrial DNA across the geographic range of three out of four subspecies. Our results suggest that far-northern populations of O. g. lasiotis potentially represent an independently evolving lineage, but populations assigned to O. g. garnettii from Zanzibar Island and of O. g panganiensis from mainland Tanzania do not constitute two independent lineages. A dated phylogeny suggests that this northern clade diverged from all remaining samples approximately 4 Mya. Such old divergence age is in line with the split between many galagid species. This northern lineage could potentially represent an incipient species; however, there is not yet enough evidence to support a new taxonomic status for this unique mitochondrial group.