Approximation Algorithms for Clustering and Vehicle Routing

聚类和车辆路径的近似算法

• 批准号: RGPIN-2020-04043
• 负责人: Friggstad, Zachary
• 金额: $ 3.5万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=739642
• 关键词: Approximation; Algorithms; Clustering; Vehicle; Routing

Challenging optimization problems appear in many aspects of our society. For example, how should one schedule a fleet of vehicles to deliver thousands of packages in a day? How can we allocate hundreds of courses to various classrooms on campus while respecting a number of auxiliary constraints such as avoiding overlap between courses that are likely to be taken by the same students? How can we place components on a circuit board and connect them via wires while keeping the entire board as small as possible? Further challenges are easy to find in many other subjects: airport scheduling, data clustering, medical testing, mobile aid station placement, etc. Poor solutions can sometimes result in millions of dollars lost. Many of these problems are so large that we rely on algorithms to find their solutions. Unfortunately, the models we use to address these problems are provably hard; being able to efficiently find the optimum solution with an algorithm would yield an unexpected resolution to the fundamental P vs. NP problem in computing science. Despite this barrier, we still need to address these important problems. To cope with this difficulty, I design approximation algorithms for hard optimization problems. These are efficient algorithms that find near-optimal solutions. This is not a mere heuristic statement, such algorithms also come with a proof that the cost of solutions they produce are within some bounded error of the best possible solution's cost. With such an approach, we have confidence that the algorithms we design are guaranteed to perform well when used on data sets that may not have been considered when the algorithm was designed. This research focuses on two very common classes of problems: data clustering and vehicle routing. Data clustering is used to support the results in hundreds of research papers every year with applications stemming from data mining, machine learning, image processing, statistical analysis, big data visualization, etc. Other applications include operations research, for example when we want to place distribution centres to serve clients in a cost-effective manner. The algorithms that are used in practice typically work well, but are known to perform poorly in certain instances. My work will focus on designing improved approximations for clustering models that, provably, work well in every instance of the problem. I am also interested in articulating, from a fundamental point of view, why certain algorithms perform better than their worst-case guarantees on "real-world" data. My work on vehicle routing will focus on improving service times and operating costs by designing improved approximations for some fundamental models beyond the well-known Traveling Salesperson Problem. Further, how do solutions to these models relate? How can solving one vehicle routing problem provide better approximation algorithms for other models?

具有挑战性的优化问题出现在我们社会的许多方面。例如，应该如何安排车队在一天内运送数千个包裹？我们如何才能将数百门课程分配到校园的各个教室，同时尊重一些辅助约束，例如避免同一学生可能修读的课程之间的重叠？我们如何将元件放置在电路板上并通过电线连接它们，同时保持整个电路板尽可能小？在许多其他主题中很容易发现进一步的挑战：机场调度、数据集群、医学测试、移动急救站安置等。糟糕的解决方案有时会导致数百万美元的损失。其中许多问题都非常大，以至于我们依靠算法来找到解决方案。不幸的是，我们用来解决这些问题的模型被证明是很难的。能够通过算法有效地找到最佳解决方案将为计算科学中的基本 P 与 NP 问题带来意想不到的解决方案。尽管存在这一障碍，我们仍然需要解决这些重要问题。为了解决这个困难，我设计了硬优化问题的近似算法。这些是找到接近最佳解决方案的有效算法。这不仅仅是一个启发式的陈述，此类算法还提供了一个证明，证明它们产生的解决方案的成本在最佳可能解决方案成本的某个有限误差范围内。通过这种方法，我们有信心我们设计的算法在用于设计算法时可能没有考虑到的数据集时能够保证表现良好。这项研究重点关注两类非常常见的问题：数据聚类和车辆路径。数据聚类每年用于支持数百篇研究论文的结果，其应用源于数据挖掘、机器学习、图像处理、统计分析、大数据可视化等。其他应用包括运筹学，例如当我们想要放置配送中心以经济高效的方式为客户提供服务。实践中使用的算法通常运行良好，但在某些情况下表现不佳。我的工作将集中于设计改进的聚类模型近似值，事实证明，这些模型在问题的每个实例中都能很好地工作。我也有兴趣从基本角度阐明为什么某些算法在“真实世界”数据上的表现优于其最坏情况的保证。我在车辆路径方面的工作将侧重于通过为众所周知的旅行推销员问题之外的一些基本模型设计改进的近似值来改善服务时间和运营成本。此外，这些模型的解决方案有何关联？解决一个车辆路径问题如何为其他模型提供更好的近似算法？