Lecture 1: Approximation Algorithms, Approximation Ratios, Gap Problems

To date, thousands of natural optimization problems have been shown to be NP-hard [6, 13]. Designing approximation algorithms [4, 17, 21] has become a standard path to attack these problems. For some problem, however, it is even NP-hard to approximate the optimal solution to within a certain ratio. The TRAVELING SALESMAN PROBLEM (TSP), for instance, has no approximation algorithm, since finding a feasible solution (the HAMILTONIAN CIRCUIT problem) is already NP-hard. Until 1990, few inapproximability results were known. The reader is referred to recent surveys by Feige [10] and Trevisan [20] for good discussions on this point and related history. To prove a typical inapproximability result such as MAX-CLIQUE is not approximable to within some ratio r (unless P=NP), a natural direction is to find a reduction from some NP-complete problem, say 3SAT, to MAXCLIQUE which satisfies the following properties: