Suppose I had a complete bipartite graph with edges each given some numerical "cost" value. Is there a way to select a subset of those edges such that each vertex on each side of the graph is mapped to each vertex on the other (one to one) and the total "costs" is maximized (or minimized)?

Has anyone ever formulated something equivalent to this?

avertex on the other? (noteach. The way you wrote it it looks like you're searching for a bi-clique). If that's the case, you should look at 'maximum weighted matching' which has a polynomial time algorithm. – Or Zuk May 14 '11 at 11:56