Transshipment, also known under the names of earth mover's distance, uncapacitated min-cost flow, or Wasserstein's metric, is an important and well-studied problem that asks to find a flow of minimum cost that routes a general demand vector. Adding to its importance, recent advancements in our understanding of algorithms for transshipment have led to breakthroughs for the fundamental problem of computing shortest paths. Specifically, the recent near-optimal $(1+\varepsilon)$-approximate single-source shortest path algorithms in the parallel and distributed settings crucially solve transshipment as a central step of their approach. The key property that differentiates transshipment from other similar problems like shortest path is the so-called \emph{boosting}: one can boost a (bad) approximate solution to a near-optimal $(1 + \varepsilon)$-approximate solution. This conceptually reduces the problem to finding an approximate solution. However, not all approximations can be boosted -- there have been several proposed approaches that were shown to be susceptible to boosting, and a few others where boosting was left as an open question. The main takeaway of our paper is that any black-box $\alpha$-approximate transshipment solver that computes a \emph{dual} solution can be boosted to an $(1 + \varepsilon)$-approximate solver. Moreover, we significantly simplify and decouple previous approaches to transshipment (in sequential, parallel, and distributed settings) by showing all of them (implicitly) obtain approximate dual solutions. Our analysis is very simple and relies only on the well-known multiplicative weights framework. Furthermore, to keep the paper completely self-contained, we provide a new (and arguably much simpler) analysis of multiplicative weights that leverages well-known optimization tools to bypass the ad-hoc calculations used in the standard analyses.

翻译：转运问题（又称推土机距离、无容量限制最小费用流或Wasserstein度量）是一个重要且被广泛研究的问题，旨在找到一条满足一般需求向量的最小费用流。更显其重要性的是，近年来对转运算法理解的进展，已为计算最短路径这一基本问题带来了突破。具体而言，近期在并行和分布式环境下提出的近乎最优的$(1+\varepsilon)$-近似单源最短路径算法，其核心步骤正是求解转运问题。转运问题区别于最短路径等其他类似问题的关键特性是所谓的“提升”：人们可以将一个（较差的）近似解提升为近乎最优的$(1+\varepsilon)$-近似解。这在概念上将问题简化为寻找一个近似解。然而，并非所有近似解都能被提升——已有若干方法被证明可被提升，而另一些方法的提升可能性则被作为开放问题遗留。本文的主要结论是：任何计算对偶解的$\alpha$-近似转运求解器（作为黑盒）都可以被提升为$(1+\varepsilon)$-近似求解器。此外，我们通过证明所有现有方法（在顺序、并行及分布式环境下）均（隐式地）获得了近似对偶解，从而显著简化并解耦了先前对转运问题的处理方法。我们的分析极为简洁，仅依赖于著名的乘法权重框架。进一步地，为保持论文的自洽性，我们提供了乘法权重新颖（且无疑更简单）的分析，该分析借助优化工具绕过了标准分析中使用的特设计算。