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: $ 3.68万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=690232
• 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。这将通过提供以下问题的定性和定量答案来实现：（i）“某些特征（特定问题实例的属性或正在解决的计算框架的属性）以什么方式和在多大程度上有助于解决特定家庭问题的内在困难？”及其补充（ii）“我们可以以什么方式以及在多大程度上利用某些自然发生的特征或约束，或修改计算框架，为这些问题的实际实例提供更有效的解决方案？”****许多我们建议解决的问题涉及运动中的物体，其有效解决方案需要清楚地理解连续几何约束与物体和环境的离散组合属性之间的相互作用。我们将在各种运动/环境约束下解决简单机器人的路径规划问题，并提供具体示例，这些具体问题因其在各种应用中发挥的基本作用而有所区别。因此，它们为在上述第二个问题上取得有意义的进展提供了肥沃的土壤。然而，这些问题也可以而且应该被视为更广泛的类似结构问题的代表。从这个意义上说，从更理论的角度来看，它们促进了第一个问题的进展。最终，我们的成功应该根据设计和分析高效算法和数据结构的新技术来衡量，以及识别适用于最普遍可能环境的固有复杂性限制。******我们的方法包括（i）仔细选择问题（牢记我们的实际和理论影响的双重目标），（ii）一方面识别和利用算法和数据结构设计任务之间的关键相互作用，然后制定另一个现实计算模型的复杂性下限，以及 (iii)对实际问题的敏感性（包括高效的、通常是自适应的算法以及稳健的实现）以及理论考虑（包括渐近最优性、硬度和完整性结果以及其他与模型相关的问题）。******要有效实现该研究计划需要考虑和利用替代计算模型（包括顺序、并行、分布式和随机模型），注意结构和资源权衡，以及在设计和分析中利用/增强其他已建立的技术和结果算法和高级数据结构。************