Summary: On-line Bipartite Matching Made Simple
We examine the classic on-line bipartite matching problem studied by Karp, Vazirani, and
Vazirani  and provide a simple proof of their result that the Ranking algorithm for this
problem achieves a competitive ratio of 1 - 1/e.
Introduced in 1990 by Karp, Vazirani, and Vazirani , on-line bipartite matching was one of the
first problems to receive the attention of competitive analysis. The input to the problem is a
bipartite graph G = (U, V, E), in which the vertices in U arrive in an on-line fashion and the edges
incident to each vertex u U are revealed when u arrives. When this happens, the algorithm may
match u to a previously unmatched adjacent vertex in V , if there is one. Such a decision, once
made, is irrevocable. The objective is to maximize the size of the resulting matching.
In their paper, Karp et al. first show that the competitiveness of the problem is trivial in
the deterministic case. Any algorithm that always matches a vertex in U if a match is possible
constructs a maximal matching, and therefore such an algorithm has a competitive ratio of 1/2.
On the other hand, given any deterministic algorithm, it is easy to construct an input (in which
each vertex u U has one or two neighbors in V ) that forces that algorithm to find a matching of
size no greater than half of the optimum.