Computational complexity of geometric and combinatorial problems

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The objective of the proposed research is to further our understanding of the inherent computational complexity of a number of fundamental combinatorial and geometric problems. Our main goal is to provide answers to two complementary questions: (i) "In what ways and to what extent do certain structural features (attributes of specific problem instances) contribute to the difficulty of solving problems in a particular family?" and (ii) "In what ways and to what extent can certain naturally occurring features or constraints be exploited to provide more efficient solutions for problem instances that exhibit these features?" The problems that we propose to address (like those addressed in my previous research) are distinguished by the fundamental role that they play in a variety of applications, including robotic motion planning and optimization, facility and sensor location, efficient schemes for broadcasting streamed information and optimization of log milling. This, of course, provides significant practical motivation for progress on the second of the questions above. However, the problems are also chosen for their ability to serve as representatives of broader families of similarly structured problems. For this reason they motivate, from a more theoretical standpoint, progress on the first question. Ultimately, our success should be measured in terms of (i) the development of new general techniques for the design and analysis of efficient algorithms and data structures, and (ii) the elucidation of inherent complexity limitations that apply in the broadest possible computational context.