A lazy approach to on-line bipartite matching

TL;DR

This paper introduces a lazy matching algorithm for online bipartite matching that surpasses the traditional 1/2 competitive ratio barrier, achieving approximately 0.588, and is proven to be optimal.

Contribution

It proposes the first optimal deterministic lazy matching algorithm for online bipartite matching with a competitive ratio of about 0.588.

Findings

Lazy matching breaks the 1/2 competitive ratio barrier.

The optimal lazy algorithm has a competitive ratio of approximately 0.588.

The approach is proven to be optimal among deterministic algorithms.

Abstract

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 $1-\pi/\cosh(\frac{\sqrt{3}}{2}\pi)\approx 0.588$. 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.