Sufficient conditions for the optimality of the greedy algorithm in greedoids

Abstract

Greedy algorithms are among the most elementary ones in theoretical computer science and understanding the conditions under which they yield an optimum solution is a widely studied problem. Greedoids were introduced by Korte and Lovász at the beginning of the 1980s as a generalization of matroids. One of the basic motivations of the notion was to extend the theoretical background behind greedy algorithms beyond the well-known results on matroids. Indeed, many well-known algorithms of a greedy nature that cannot be interpreted in a matroid-theoretical context are special cases of the greedy algorithm on greedoids. Although this algorithm turns out to be optimal in surprisingly many cases, no general theorem is known that explains this phenomenon in all these cases. Furthermore, certain claims regarding this question that were made in the original works of Korte and Lovász turned out to be false only most recently. The aim of this paper is to revisit and straighten out this question: we summarize recent progress and we also prove new results in this field. In particular, we generalize a result of Korte and Lovász and thus we obtain a sufficient condition for the optimality of the greedy algorithm that covers a much wider range of known applications than the original one.

A Note on Ultrametric Spaces, Minimum Spanning Trees and the Topological Distance Algorithm

“…Remark 4. A subtlety might be worthwhile/important to note: It is claimed in [9] that the general proof of the optimality of the greedy algorithm for general greedoids, i.e., Theorem XI.1.3 of [4] contains a subtle error. However, our proofs are not affected by this argument, as Kruskal's algorithm applies to matroids, as pointed out therein, and Prim's algorithm-known to be valid for a long time anyway-is covered by Theorem XI.2.2 of [4] (see Theorem 9 of [9]).…”

New polyhedral and algorithmic results on greedoids

“…In [10] a variant of the greedy algorithm on interval greedoids that "looks two step ahead" is defined and a necessary and sufficient condition for its optimality on linear objective functions is derived. As for general (that is, not necessarily linear) and possibly order-dependent objective functions a generalization of Theorem 2 was most recently given in [15] that, among other applications, completely covers Example 3 (also for undirected graphs).…”