Lec . 1 : Approximation Algorithms for NP - hard problems

In this course, we will be studying, as the title suggests, the approximability and inapproximability (limits of approximability) of different combinatorial optimization problems. All the problems we will be looking at will be ones that lack efficient algorithms and in particular will be NP-hard problems. The last two-three decades has seen remarkable progress in approximation algorithms for several of these NP-hard problems. The theory of NPcompleteness provides a satisfactory theory of algorithmic hardness in many ways, however it is unable to explain the vastly different approximabilities of different NP-hard problems. Since the early 90’s, work on probabilistic proof systems have shed light on the limitations to approximation algorithms. We will study in detail this connection between proof systems and (in)approximability and spend considerable time on the construction of such proof systems. In the first few lectures, we will discuss several approximation algorithms before we proceed to the topic of the course, “limits of approximation algorithm”. This course will be a fleshed out version of a recent DIMACS tutorial by the same name [HC09].