This project involves the design and analysis of algorithms for combinatorial problems using techniques from linear algebra and convex optimization. The project bridges pure mathematics and data science, allowing new applications of mathematical ideas to the practice of computing, and from the activities on dissemination, exposition, outreach and mentoring. The project will create open-access lecture notes, surveys and blog posts, making highly technical results accessible to a broader audience. This project will play a key role in the training of graduate students, including two students belonging to underrepresented groups in computer science, and in the design of a new graduate course.
The project involves novel approaches to fundamental problems, such as certifying the unsatisfiability of random constraint satisfaction problems, efficiently certifying properties of sparse random graphs and sparse random matrices, understanding the power of sub-exponential size relaxations in the sum-of-squares hierarchies, developing new construction of graph sparsifiers and finding new ways to analyze certain probabilistic distributed processes. Some of the problems in the scope of this project are not believed to admit algorithms that perform correctly and efficiently on all inputs. For this reason, the project will focus on: (a) algorithms whose running time scale "sub-exponentially" and that outperform brute-force combinatorial search, and (b) algorithms that may perform poorly on a few inputs but that perform well on average on random inputs. The second goal depends on how the inputs are distributed, and a key focus of this project is to generalize past results that apply to certain specific distributions to broader classes of distributions.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
