How fundamental are matroids and greedoids in algorithm design?

Question

Initially, matroids were introduced to generalize the notions of linear independence of a collection of subsets $E$ over some ground set $I$. Certain problems that contain this structure permit greedy algorithms to find optimal solutions. The concept of greedoids was later introduced to generalize this structure to capture more problems that allow for optimal solutions to be found by greedy methods.

How often do these structures arise in algorithm design?

Furthermore, more often than not a greedy algorithm will not be able to fully capture what is necessary to find optimal solutions, but may still find very good approximate solutions (Bin Packing, for example). Given that, is there a way to measure how "close" a problem is to a greedoid or matroid?

Answer 1

It's difficult to answer the question "how often". But as with all "underlying structures" the benefit comes from recognizing that the underlying problem one is trying to solve has a matroid (or greedoid) structure. It's not just matroid problems. The matroid intersection problem has a specific model (bipartite matching).

Nick Harvey did his Ph.D thesis fairly recently on algorithms for matroid problems and also looked at submodular function optimization (which generalizes matroid problems). Reading the introduction and background to the thesis might be helpful.