多边际最优运输问题的硬度结果

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
Jason M. Altschuler, Enric Boix-Adserà
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引用次数: 23

摘要

多边际最优运输(MOT)问题是固定边际联合概率分布上的线性规划问题。许多应用中的一个关键问题是求解MOT的复杂性:线性规划在边际数量k及其支持大小n上具有指数大小。最近的一系列工作表明,对于具有poly(n,k)大小隐式表示的某些成本族,MOT是poly(n,k)时间可解的。然而,目前还不清楚这条算法研究路线还能包含多少成本。为了了解这些基本的局限性,本文开始了MOT的难处理结果的研究。我们的主要技术贡献是开发一个工具包,用于证明np -硬度和MOT问题的不可逼近性结果。该工具包将证明MOT问题的难解性降低到证明更易于处理的离散优化问题的难解性。我们通过使用它来建立文献中研究的许多MOT问题的难解性来演示该工具包,这些问题抵制了以前的算法努力。例如,我们提供的证据表明,排斥成本使MOT难以解决,通过显示几个这样的感兴趣的问题是np -难以解决,甚至近似。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Hardness results for Multimarginal Optimal Transport problems

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. This toolkit reduces proving intractability of MOT problems to proving intractability of more amenable discrete optimization 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.

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来源期刊
Discrete Optimization
Discrete Optimization 管理科学-应用数学
CiteScore
2.10
自引率
9.10%
发文量
30
审稿时长
>12 weeks
期刊介绍: Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.
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