The Greedy Algorithm

In this chapter we study a method for optimizing over certain set systems, the so-called greedy algorithm. More precisely, it is used for maximizing a weight function on so-called independence systems, the classical instance being the system of spanning forests of a graph. The greedy strategy is rather short-sighted: one always selects the element which seems best at the moment. In other words, among all the admissible elements, one chooses that with maximal weight and adds it to the solution being constructed. In general, this simple-minded strategy will not work, but for a certain class of structures playing a very important part in combinatorial optimization, the so-called matroids, it indeed leads to optimal solutions. Actually, matroids may even be characterized by the fact that the greedy algorithm works for them, but there are other possible definitions. We shall look at various other characterizations of matroids and also consider the notion of matroid duality. Following this, we will consider the greedy algorithm as an approximation method for maximization on independence systems which are not matroids. We examine the efficiency of this approach, that is, we derive bounds for the ratio between the solution given by the greedy algorithm and the optimal solution. We also look at the problem of minimization on independence systems. Finally, in the last section, we discuss some further generalizations of matroids and their relationship to the greedy algorithm.

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