Mathematical programming is a powerful tool for attacking combinatorial problems. One transforms a discrete task into a related continuous one by casting it as optimization over a convex body. Linear and semi-definite programming (LP and SDP) form important special cases and are central tools in the theory and practice of combinatorial optimization. These approaches have achieved spectacular success in computing approximately optimal solutions for problems where finding exact solutions is computationally intractable.
While there are very strong bounds known on the efficacy of particular families of relaxations, it remains possible that adding a small number of variables or constraints could lead to drastically improved solutions. We propose the development of a theory to unconditionally capture the power of LPs and SDPs without any complexity-theoretic assumptions. Our approach has the potential to show something remarkable: For many well-known problems, the basic LP or SDP is optimal among a very large class of algorithms. More concretely, we suggest a method that could rigorously characterize the power of polynomial-size LPs and SDPs for a variety of combinatorial optimization tasks. This involves deep issues at the intersection of many areas of mathematics and computer science, with the ultimate goal of significantly extending our understanding of efficient computation.
Mathematical programming is of major importance to many fields---this is especially true for computer science and operations research. These methods have also seen dramatically increasing use in the analysis of "big data" from across the scientific spectrum. From a different perspective, LPs and SDPs can be thought of as rich proof systems, and characterizing their power is a basic problem in the theory of proof complexity. Thus the outcomes of the proposed research are of interest to a broad community of scientists, mathematicians, and practitioners.
