Some of you asked for intuition about matroids: they seem so abstract, how to think about them? Whenever I have to prove a fact about matroids, I think about how I’d prove the fact for graphic matroids. And often (though alas, not always) that proof translates seamlessly to general matroids.
Recall that a graphic matroid is constructed as follows: take a graph. Its edges are the elements of the matroid. A set of elements (edges) is independent if they do not induce a cycle (in the graph-theory sense), i.e., if they form a forest. Now you can check that this definition satisfies the conditions for being a matroid. The bases of the matroid are spanning trees of G.
So on HW3, the first part asked: suppose I have two spanning trees on a graph G, a red one and a blue one. Show that there exists a bijection F between the red and blue edges so that replacing any red edge e by its matched blue edge F(e) gives a spanning tree. The proof now seems well within reach. And the arguments to prove it (adding an element/edge forms a cycle, etc) all generalize to matroids.
Why are we studying matroids? They form a convenient abstraction for many combinatorial structures you often encounter. E.g., spanning trees, linearly-independent sets of vectors, matchable vertices in a bipartite graph, sets of vertices that can routed to a sink in some vertex-disjoint way, all are matroids. E.g., perfect matchings in bipartite graphs can be phrased as finding a common base of two matroids defined on the same ground set. A famous result of Edmonds shows that this matroid intersection problem can be solved for any two matroids. Also, since matroids characterize the greedy algorithm (in a sense you saw in HW1), the fact that the greedy algorithm works for some problem often indicates there may be some deeper structure lurking within, that we can exploit.
I was hoping to cover matroids in a lecture later in the course (let’s see how things go), but here is a course by Jan Vondrak that covers many polyhedral aspects of matroids.