SIAM Review, Volume 63, Issue 1, Page 121-121, January 2021.
The SIGEST article in this issue is “What Tropical Geometry Tells Us about the Complexity of Linear Programming,” by Xavier Allamigeon, Pascal Benchimol, Stéphane Gaubert, and Michael Joswig. Linear programming means optimizing a linear objective function with respect to linear equality constraints and linear inequality constraints. This remains a pervasive task in modern society, where decisions must be made while keeping control of multiple factors, such as staff time, raw materials, energy, financial outlay, or carbon footprint. Linear programs also arise as subtasks for more general problems, often forming computational bottlenecks. The search for linear programming algorithms with good worst-case complexity was boosted by the work of Khachiyan (1979) and Karmarker (1984), which brought interior point methods to the fore. In this SIGEST article, the authors construct a negative result: they define a linear program for which a widely used class of interior point methods has complexity that is exponential in the number of variables (Theorem A). This leads to a counterexample for the continuous analogue of the Hirsch conjecture, proposed by Deza, Terlaky, and Zinchenko in 2009. The authors' proofs use the tools of tropical geometry, a world where addition is replaced by maximization and multiplication is replaced by standard addition. Intuitively, this approach is likely to have value in scheduling-type problems because (a) for two activities that may be performed concurrently, the time required is the maximum of the individual times, and (b) for two activities that must take place consecutively, the time required is the sum of the individual times. The original version of this article appeared in the SIAM Journal on Applied Algebra and Geometry in 2018. In preparing this SIGEST version, the authors have added new material to sections 1 and 2 in order to increase accessibility, and in section 9 they have included a discussion of further work in this area and relevant open problems.

中文翻译：

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SIGEST

SIAM 评论，第 63 卷，第 1 期，第 121-121 页，2021 年 1 月。
本期 SIGEST 文章是 Xavier Allamigeon、Pascal Benchimol、Stéphane Gaubert 和 Michael Joswig 撰写的“热带几何告诉我们线性规划的复杂性”。线性规划意味着针对线性等式约束和线性不等式约束优化线性目标函数。在现代社会，这仍然是一项普遍的任务，必须在做出决策的同时控制多种因素，例如员工时间、原材料、能源、财务支出或碳足迹。线性程序也作为更一般问题的子任务出现，通常形成计算瓶颈。Khachiyan (1979) 和 Karmarker (1984) 的工作推动了对具有良好的最坏情况复杂度的线性规划算法的搜索，它们使内点方法脱颖而出。在这篇 SIGEST 文章中，作者构建了一个否定结果：他们定义了一个线性程序，其中广泛使用的一类内点方法具有变量数量呈指数级的复杂性（定理 A）。这引出了 Deza、Terlaky 和 ​​Zinchenko 在 2009 年提出的 Hirsch 猜想的连续类比的反例。 作者的证明使用热带几何的工具，在一个世界中，加法被最大化取代，乘法被标准取代添加。直觉上，这种方法可能对调度类型的问题有价值，因为（a）对于可能同时执行的两个活动，所需的时间是单个时间中的最大值，以及（b）对于两个必须连续进行的活动，所需时间是各个时间的总和。