We present a new approach, called a lazy matching, to the problem of on-line
matching on bipartite graphs. Imagine that one side of a graph is given and the
vertices of the other side are arriving on-line. Originally, incoming vertex is
either irrevocably matched to an another element or stays forever unmatched. A
lazy algorithm is allowed to match a new vertex to a group of elements
(possibly empty) and afterwords, forced against next vertices, may give up
parts of the group. The restriction is that all the time each element is in at
most one group. We present an optimal lazy algorithm (deterministic) and prove
that its competitive ratio equals 1βΟ/cosh(23ββΟ)β0.588. The lazy approach allows us 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