Online Matching: Haste makes Waste!

This paper studies a new online problem, referred to as \emph{min-cost perfect matching with delays (MPMD)}, defined over a finite metric space (i.e., a complete graph with positive edge weights obeying the triangle inequality) $\mathcal{M}$ that is known to the algorithm in advance. Requests arrive in a continuous time online fashion at the points of $\mathcal{M}$ and should be served by matching them to each other. The algorithm is allowed to delay its request matching commitments, but this does not come for free: the total cost of the algorithm is the sum of metric distances between matched requests \emph{plus} the sum of times each request waited since it arrived until it was matched. A randomized online MPMD algorithm is presented whose competitive ratio is $O (\log^{2} n + \log \Delta)$, where $n$ is the number of points in $\mathcal{M}$ and $\Delta$ is its aspect ratio. The analysis is based on a machinery developed in the context of a new stochastic process that can be viewed as two interleaved Poisson processes; surprisingly, this new process captures precisely the behavior of our algorithm. A related problem in which the algorithm is allowed to clear any unmatched request at a fixed penalty is also addressed. It is suggested that the MPMD problem is merely the tip of the iceberg for a general framework of online problems with delayed service that captures many more natural problems.