Computational Complexity of Geometric and Combinatorial Problems

几何和组合问题的计算复杂性

• 批准号: RGPIN-2016-04274
• 负责人: Kirkpatrick, David
• 金额: $ 7.36万
• 依托单位: University of British Columbia
• 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=749843
• 关键词: Computational; Complexity; Geometric; Combinatorial; Problems

The primary objective of the proposed research is to further our understanding of the inherent computational complexity of a number of fundamental combinatorial and geometric problems. This will be achieved by providing qualitative and quantitative answers to the questions: (i) "In what ways and to what extent do certain features (attributes of specific problem instances, or of the computational framework in which they are being addressed) contribute to the intrinsic difficulty of solving problems in a particular family?" and its complement (ii) "In what ways and to what extent can we exploit certain naturally occurring features or constraints, or modify the computational framework, to provide more efficient solutions to practical instances of these problems?"Many of the problems that we propose to address involve objects in motion, and their effective solution requires a clear understanding of the interplay between continuous geometric constraints and the discrete combinatorial attributes of the objects and environment. Specific problems that we will address path-planning problems for simple robots, under various motion/environment constraints, provide concrete examples are distinguished by the fundamental role that they play in a wide variety of applications. As such they provide fertile ground for meaningful progress on the second of the questions above. However, the problems can, and should, also be viewed as representatives of broader families of similarly structured problems. In this sense they facilitate, from a more theoretical standpoint, progress on the first question. Ultimately, our success should be measured in terms of new techniques for the design and analysis of efficient algorithms and data structures, and for the identification of inherent complexity limitations that apply in the most general possible context.Our methodology involves (i) the careful selection of problems (bearing in mind our dual objectives of practical and theoretical impact), (ii) the identification and exploitation of the critical interplay between tasks of algorithm and data structure design, on one hand, and the formulation of lower bounds on complexity for realistic models of computation on the other, and (iii) a sensitivity for practical issues (including efficient, often adaptive, algorithms together with robust implementations) as well as theoretical considerations (including asymptotic optimality, hardness and completeness results, and other model-dependent concerns).To be effectively realized this research program requires consideration and utilization of alternative models of computation (including sequential, parallel, distributed and randomized models), attention to structural and resource trade-offs, and the exploitation/enhancement of other established techniques and results in both the design and analysis of algorithms and advanced data structures.

拟议研究的主要目的是进一步了解许多基本组合和几何问题的固有计算复杂性。这将通过为问题提供定性和定量答案来实现：（i）“在什么方面和在什么方面具有某些特征（特定问题实例的属性或解决的计算框架的属性）为该问题做出了贡献）在特定家庭中解决问题的内在困难？”它的补充（ii）”“以多种方式以及在多大程度上可以利用某些自然发生的特征或约束或修改计算框架，以为这些问题的实际实例提供更有效的解决方案？”要解决涉及运动中的对象，其有效的解决方案需要清楚地了解对象和环境的连续几何约束与离散组合属性之间的相互作用。在各种运动/环境限制下，我们将解决简单机器人的路径规划问题的具体问题，提供了具体示例，这是通过它们在各种应用中扮演的基本角色来区分的。因此，它们为上述第二个问题的第二个问题提供了有意义的进步。但是，这些问题可以并且应该被视为类似结构化问题的更广泛家庭的代表。从这个意义上讲，从更理论的角度来看，它们在第一个问题上有助于进步。最终，应根据新技术的设计和分析有效算法和数据结构的设计和分析，以及确定适用于最一般可能的上下文中的固有复杂性限制。问题（考虑到我们实用和理论影响的双重目标），（ii）一方面，算法和数据结构设计之间的关键相互作用的识别和剥削，一方面，以及对现实性的复杂性的表述对另一个的计算模型，以及（iii）对实际问题（包括有效的，通常是适应性的算法以及可靠的实现）以及理论上的考虑（包括渐近的最佳性，硬度和完整性结果，以及其他依赖模型依赖性的关注点）的敏感性）。要有效地实现该研究计划，需要考虑和利用替代的计算模型（包括顺序，并行，分布式模型和随机模型），对结构和资源折衷的关注以及对其他既定技术和结果的开发/增强在算法和高级数据结构的设计和分析中。