Tighter Bounds for Online Bipartite Matching

We study the online bipartite matching problem, introduced by Karp, Vazirani and Vazirani [1990]. For bipartite graphs with matchings of size n, it is known that the Ranking randomized algorithm matches at least $$(1 - \frac{1}{e})n$$ edges in expectation. It is also known that no online algorithm matches more than $$(1 - \frac{1}{e})n + O(1)$$ edges in expectation, when the input is chosen from a certain distribution that we refer to as $$D:n$$. This upper bound also applies to fractional matchings. We review the known proofs for this last statement. In passing we observe that the O(1) additive term (in the upper bound for fractional matching) is $$\frac{1}{2} - \frac{1}{2e} + O(\frac{1}{n})$$, and that this term is tight: the online algorithm known as Balance indeed produces a fractional matching of this size. We provide a new proof that exactly characterizes the expected cardinality of the (integral) matching produced by Ranking when the input graph comes from the support of $$D:n$$. This expectation turns out to be $$(1 - \frac{1}{e})n + 1 - \frac{2}{e} + O(\frac{1}{n!})$$, and serves as an upper bound on the performance ratio of any online (integral) matching algorithm.