The set cover problem plays the same role in approximation algorithms that the maximum matching problem played in exact algorithms — as a problem whose study led to the development of fundamental techniques for the entire field. For our purpose this problem is particularly useful, since it offers a very simple setting in which many of the basic algorithm design techniques can be explained with great ease. In this chapter, we will cover two combinatorial techniques: the fundamental greedy technique and the technique of layering. In Part II we will explain both the basic LP-based techniques of rounding and the primal-dual schema using this problem. Because of its generality, the set cover problem has wide applicability, sometimes even in unexpected ways. In this chapter we will illustrate such an application — to the shortest superstring problem (see Chapter 7 for an improved algorithm for the latter problem).
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