Computational Complexity of Geometric and Combinatorial Problems

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