This study examines the time complexities of the unbalanced optimal transport problems from an algorithmic perspective for the first time. We reveal which problems in unbalanced optimal transport can/cannot be solved efficiently. Specifically, we prove that the Kantorovich Rubinstein distance and optimal partial transport in the Euclidean metric cannot be computed in strongly subquadratic time under the strong exponential time hypothesis. Then, we propose an algorithm that solves a more general unbalanced optimal transport problem exactly in quasi-linear time on a tree metric. The proposed algorithm processes a tree with one million nodes in less than one second. Our analysis forms a foundation for the theoretical study of unbalanced optimal transport algorithms and opens the door to the applications of unbalanced optimal transport to million-scale datasets.

翻译：本研究首次从算法角度审查了不平衡的最佳运输问题的时间复杂性。 我们首次从算法角度揭示了不平衡的最佳运输不均/无法有效解决的一些问题。 具体地说, 我们证明坎托罗维奇·鲁宾斯坦的距离和欧clidean 度量的优化部分运输不能在强烈指数时间假设下以强烈的亚水边时间计算。 然后, 我们提出了一个算法, 解决更普遍的不平衡最佳运输问题, 恰好在树的准线性时间。 提议的算法处理一棵树, 其100万个节点在不到一秒的时间里。 我们的分析构成了对不平衡的最佳运输算法的理论研究的基础, 并为将不平衡的最佳运输应用到百万尺度的数据集打开了大门。