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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=750500
• 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?

最优化问题出现在社会生活的各个方面。例如，如何安排一个车队在一天内运送数千个包裹？我们如何才能在校园内的各个教室分配数百门课程，同时又尊重一些辅助限制，例如避免同一学生可能选修的课程之间的重叠？我们如何在电路板上放置元件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，数据聚类，医疗测试，移动的援助站的放置等。这些问题中有许多是如此之大，以至于我们依赖算法来找到它们的解决方案。不幸的是，我们用来解决这些问题的模型被证明是困难的;能够有效地找到最佳解决方案的算法将产生一个意想不到的解决方案的基本P与NP问题在计算科学。尽管存在这一障碍，我们仍然需要解决这些重要问题。为了科普这一问题，我设计了求解困难优化问题的近似算法。这些都是找到接近最优解的有效算法。这不仅仅是一个启发式的陈述，这样的算法还证明了它们产生的解决方案的成本在最佳可能解决方案的成本的有界误差内。有了这样的方法，我们有信心，我们设计的算法可以保证在设计算法时可能没有考虑的数据集上执行良好。本研究集中在两个非常常见的问题：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，其应用来自数据挖掘，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营研究，例如当我们想要以具有成本效益的方式为客户提供配送中心时。在实践中使用的算法通常工作良好，但已知在某些情况下表现不佳。我的工作将集中在为聚类模型设计改进的近似，可以证明，在问题的每一个实例中都能很好地工作。我也有兴趣从基本的角度阐明，为什么某些算法在“真实世界”数据上的表现比它们在最坏情况下的保证更好。我在车辆路径方面的工作将集中在通过为一些基本模型设计改进的近似来改善服务时间和运营成本，这些模型超出了著名的旅行推销员问题。此外，这些模型的解决方案如何相互关联？如何解决一个车辆路径问题为其他模型提供更好的近似算法？