27.1 Framework

So far in this course, we have been proving upper bounds on the approximation factors achievable for certain NP -hard problems by giving approximation algorithms for them. In this lecture, we shift gears and prove lower bounds for some of these problems, under the assumption that P 6= NP . For some problems, essentially matching upper and lower bounds are known, indicating that the approximability properties of these problems are well-understood. We show that (nonmetric) TSP cannot be approximated within any factor. We also show that the Edge Disjoint Paths problem is NP -hard to approximate within a factor of O(m1/2− ) for all > 0, essentially matching the O( √ m) algorithm obtained in a previous lecture. Finally, we give a bootstrapping argument showing that if the Clique problem cannot be approximated within some constant factor, then it cannot be approximated within any constant factor.