Algorithms and Resource Requirements for Fundamental Problems (Thesis Proposal)

We establish more efficient methods for solving certain classes of NP-hard problems exactly, as well as methods for proving limitations on how quickly the same problems can be solved. For example, in the Max-Cut problem, one is given a graph G = (V,E) and integer K, and one wishes to determine if G has a subset of vertices such that the number of edges leaving the subset is at least K. The trivial algorithm for Max-Cut runs in roughly O(poly(n) · 2n) time, where n is the number of vertices in G. Prior to our work, no better algorithm was known for the general case. Our results imply that Max-Cut can be solved in O( √ 3 n ) = O(1.74n) time and space, yet Max-Cut cannot be solved in O(n √ ) time and n space. Further study includes finding faster algorithms and proving larger limitations for these problems, as well as extending our techniques to counting versions of NP problems. We have already extended some of our algorithms in this manner, and are currently working on how limitations can be proved.