Discrete Optimal Transport with Independent Marginals is #P-Hard
研究了两个K维离散随机向量之间Wasserstein距离的计算复杂性，证明即使第一个向量各分量独立且服从均匀伯努利分布、第二个向量仅有两个原子，该问题也是#P难的，并提出了一个伪多项式时间的动态规划近似算法。
We study the computational complexity of the optimal transport problem that evaluates the Wasser- stein distance between the distributions of two K-dimensional discrete random vectors. The best known algorithms for this problem run in polynomial time in the maximum of the number of atoms of the two distributions. However, if the components of either random vector are independent, then this number can be exponential in K even though the size of the problem description scales linearly with K. We prove that the described optimal transport problem is #P-hard even if all components of the first random vector are independent uniform Bernoulli random variables, while the second random vector has merely two atoms, and even if only approximate solutions are sought. We also develop a dynamic programming-type algorithm that approximates the Wasserstein distance in pseudo-polynomial time when the components of the first random vector follow arbitrary independent discrete distributions, and we identify special problem instances that can be solved exactly in strongly polynomial time.