Multimarginal Optimal Transport (MOT) is the problem of linear programming over joint probability distributions with fixed marginals. A key issue in many applications is the complexity of solving MOT: the linear program has exponential size in the number of marginals k and their support sizes n. A recent line of work has shown that MOT is poly(n,k)-time solvable for certain families of costs that have poly(n,k)-size implicit representations. However, it is unclear what further families of costs this line of algorithmic research can encompass. In order to understand these fundamental limitations, this paper initiates the study of intractability results for MOT. Our main technical contribution is developing a toolkit for proving NP-hardness and inapproximability results for MOT problems. We demonstrate this toolkit by using it to establish the intractability of a number of MOT problems studied in the literature that have resisted previous algorithmic efforts. For instance, we provide evidence that repulsive costs make MOT intractable by showing that several such problems of interest are NP-hard to solve--even approximately.
翻译:多边优化运输(MOT)是针对固定边缘点联合概率分布的线性编程问题。许多应用中的一个关键问题是解决MOT问题的复杂性:线性程序具有边际k及其支持大小的指数大小。 最近的一项工作表明,MOT是多(n,k)时间可溶于具有多(n,k)规模的隐含表示的某些成本家庭的费用家庭的多(n,k)时间可溶解。然而,尚不清楚这一算法研究线的成本可包含哪些进一步的费用家庭。为了理解这些基本限制,本文件启动了对MOT的可吸引性结果的研究。我们的主要技术贡献是开发了一个工具,用以证明NP-硬性和对MOT问题的不可利用性结果。我们用这个工具来证明在文献中研究的多(n,k)规模的隐含蓄度问题无法吸引性。例如,我们提供了证据,通过表明若干这类感兴趣的问题难以解决大约的NP-甚至。