"The main goal of this project is to lay the foundations of a ``non-polynomial time theory of approximation"" -- the Parameterized Approximation for NP-hard optimization problems. A combination that will use the salient features of Approximation Algorithms andParameterized Complexity. In the former, one relaxes the requirement of finding an optimum solution. In the latter, one relaxes the requirement of finishing in polynomial time by restricting thecombinatorial explosion in the running time to a parameter that for reasonable inputs is much smaller than the input size. This project will explore the following fundamental question:Approximation Algorithms + Parameterized Complexity=?New techniques will be developed that will simultaneously utilize the notions of relaxed time complexity and accuracy and thereby make problems for which both these approaches have failed independently, tractable. It is however conceivable that for some problems even this combined approach may not succeed. But in those situations we will glean valuable insight into the reasons for failure. In parallel to algorithmic studies, an intractability theory will bedeveloped which will provide the theoretical framework to specify the extent to which this approach might work. Thus, on one hand the project will give rise to algorithms that will have impact beyond the boundaries of computer science and on the other hand it will lead to a complexity theory that will go beyond the established notions of intractability. Both these aspects of my project are groundbreaking -- the new theory will transcend our current ideas ofefficient approximation and thereby raise the state of the art to a new level."