Recursive Greedy Methods

Greedy algorithms are often the first algorithm that one considers for various optimization problems, and ,in particular, covering problems. The idea is very simple: try to build a solution incrementally by augmenting a partial solution. In each iteration, select the “best” augmentation according to a simple criterion. The term greedy is used because the most common criterion is to select an augmentation that minimizes the ratio of “cost” to “advantage”. We refer to the cost-to-advantage ratio of an augmentation as the density of the augmentation. In the Set-Cover (SC) problem, every set S has a weight (or cost) w(S). The “advantage” of a set S with respect to a partial cover {S1, . . . , Sk} is the number of new elements covered by S, i.e., |S \ (S1 ∪ . . .∪Sk)|. In each iteration, a set with a minimum density is selected and added to the partial solution until all the elements are covered. In the SC problem, it is easy to find an augmentation with minimum density simply by re-computing the density of every set in every iteration. In this chapter we consider problems for which it is NP-hard to find an augmentation with minimum ∗Chapter 3 from: Handbook of Approximation Algorithms and Metaheuristics edited by Teofilo Gonzalez.