The project aims to develop efficient algorithms for computational problems that arise in business, engineering, and science. The project will have an impact on theoretical computer science (TCS) by improving our understanding of the nature of real-life instances and designing better algorithms -- with provable performance guarantees -- for them. Additionally, the results will be relevant to researchers in other areas of computer science, including machine learning and optimization. In particular, this project will provide researchers with new practical algorithms. Through its wide-reaching results, the project will strengthen the connection between TCS and other areas of computer science. Further, the project will be of interest to researchers in mathematics, mathematical physics, and statistics, in part because one of the key models considered in this proposal has been introduced and studied in these fields.
The PI will collaborate on this project with graduate and undergraduate students at the Toyota Technological Institute at Chicago (TTIC) and the University of Chicago. Additionally, he will invite PhD students from other universities to work on the project during summer months. The PI will incorporate the topic of this proposal in his graduate courses. Specifically, he will include introductory material on the topic in his graduate Algorithms course and more advanced material in his course on metric geometry in computer science.
Many problems that arise in business, engineering, and science are very hard in the worst case: for them, there are no "universal" efficient algorithms -- i.e., algorithms that solve all possible instances in reasonable (polynomial) time. However, real-life instances are usually considerably simpler than the most difficult ones. The goal of this project is to identify what makes real-life instances computationally tractable and to design efficient algorithms for solving them. These algorithms will solve many real-life instances of computational problems by exploiting their structural properties (at the same time, the algorithms may fail to solve the most difficult instances, which, however, almost never appear in practice).
In order to design efficient algorithms -- with provable performance guarantees -- for real-life instances of computational problems, one has to define a formal model for real-life instances. This proposal will explore existing generative and descriptive models for real-life instances of clustering and optimization problems, including semi-random stochastic block models and models based on different stability assumptions. The project will also develop new, more advanced models. The PI will design new algorithms for these models and analyze existing heuristics.
