Computational Complexity of Geometric and Combinatorial Problems

提出的研究的主要目的是进一步我们对一些基本组合和几何问题的固有计算复杂性的理解。这将通过对下列问题提供定性和定量的回答来实现：(i)“某些特征（特定问题实例的属性，或正在处理这些问题的计算框架的属性）以何种方式和在何种程度上有助于解决特定家庭问题的内在困难？”及其补充（ii）“我们可以以何种方式和在何种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？”****我们提出要解决的许多问题都涉及运动中的物体，它们的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的具体问题，提供具体的例子，这些问题由它们在各种应用中发挥的基本作用来区分。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，这些问题可以，也应该被视为更广泛的类似结构问题家族的代表。从这个意义上说，从更理论的角度来看，它们有助于在第一个问题上取得进展。最终，我们的成功应该以设计和分析有效算法和数据结构的新技术，以及在最普遍的情况下应用的固有复杂性限制的识别来衡量。******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，另一方面制定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，通常自适应，算法（以及健壮的实现）以及理论考虑（包括渐近最优性、硬度和完备性结果，以及其他模型相关的问题）。******为了有效地实现这一研究计划，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，以及在算法和高级数据结构的设计和分析中开发/增强其他已建立的技术和结果***********

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.***********