The Greedy Algorithm

Abstract

In this chapter, we look at a generalization of the Algorithm of Kruskal, the so-called Greedy Algorithm. This algorithm can be used for maximization on ‘independence systems’ (as, for example, in the case of the Algorithm of Kruskal, the system of spanning forests of a graph). However, the strategy used is rather short-sighted: we always choose the element which seems best at the moment (that is, of all the admissible elements, we choose the element whose weight is maximal) and add it to the solution we are constructing (this explains the name!). In general, this strategy does not work, but for a certain class of structures, the so-called matroids (which play an important part in Combinatorial Optimization), it indeed leads to an optimal solution. In fact, this class of structures is characterized by the fact that the Greedy Algorithm works for them, but there are other possible definitions for matroids. We will see some other characterizations of matroids and look at the notion of duality of matroids.