A lazy approach to on-line bipartite matching

We present a new approach, called a lazy matching, to the problem of on-line matching on bipartite graphs. Originally, each arriving element is irrevocably matched with some already existing one. In this paper we allow matching a new element to a group of elements (possibly decreasing other groups) with additional restrictions that every two groups have to be disjoint and decreasing a group is allowed as long as it stays not empty. We present a deterministic optimal algorithm and prove that its competitive ratio equals 1 − π/ cosh(√ 3 2 π) ≈ 0.588. The lazy approach allows to break the barrier of 1/2, which is the best competitive ratio that can be guaranteed by any deterministic algorithm in the classical on-line matching.