The Approximability of Some NP-hard Problems

An α-approximation algorithm is an algorithm guaranteed to output a solution that is within an α ratio of the optimal solution. We are interested in the following question: Given an NP-hard optimization problem, what is the best approximation guarantee that any polynomial time algorithm could achieve? We mostly focus on studying the approximability of two classes of NP-hard prob-lems: Constraint Satisfaction Problems (CSPs) and Computational Learning Problems. Our research in the field of CSPs is to show that certain Semidefinite Programming (SDP) algorithms are the optimal polynomial time approximation algorithm; our work in the learning area is to prove that tasks are inherently hard; i.e., there is no better-than-trivial algorithm for the problems. We have shown some preliminary results on the approximability of several problems from these two classes including Max-Cut, Satisfiable 3-CSPs and agnostic learning of monomials