A Deterministic Polylogarithmic Competitive Algorithm for Matching with Delays

In the online Min-cost Perfect Matching with Delays (MPMD) problem, $m$ requests in a metric space are submitted at different times by an adversary. The goal is to match all requests while (i) minimizing the sum of the distances between matched pairs as well as (ii) how long each request remained unmatched after it appeared. While there exist almost optimal algorithms when the metric space is finite and known a priori, this is not the case when the metric space is infinite or unknown. In this latter case, the best known algorithm, due to Azar and Jacob-Fanani, has competitiveness $\mathcal{O}(m^{0.59})$ which is exponentially worse than the best known lower bound of $\Omega(\log m / \log \log m)$ by Ashlagi et al. We present a $\mathcal{O}(\log^5 m)$-competitive algorithm for the MPMD problem. This algorithm is deterministic and does not need to know the metric space or $m$ in advance. This is an exponential improvement over previous results and only a polylogarithmic factor away from the lower bound.