Online k-Way Matching with Delays and the H-Metric
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In this paper, we study k-Way Min-cost Perfect Matching with Delays - the k-MPMD problem. This problem considers a metric space with n nodes. Requests arrive at these nodes in an online fashion. The task is to match these requests into sets of exactly k, such that the space and time cost of all matched requests are minimized. The notion of the space cost requires a definition of an underlying metric space that gives distances of subsets of k elements. For k>2, the task of finding a suitable metric space is at the core of our problem: We show that for some known generalizations to k=3 points, such as the 2-metric and the D-metric, there exists no competitive randomized algorithm for the 3-MPMD problem. The G-metrics are defined for 3 points and allows for a competitive algorithm for the 3-MPMD problem. For k>3 points, there exist two generalizations of the G-metrics known as n- and K-metrics. We show that neither the n-metrics nor the K-metrics can be used for the k-MPMD problem. On the positive side, we introduce the H-metrics, the first metrics to allow for a solution of the k-MPMD problem for all k. In order to devise an online algorithm for the k-MPMD problem on the H-metrics, we embed the H-metric into trees with an O(log n) distortion. Based on this embedding result, we extend the algorithm proposed by Azar et al. (2017) and achieve a competitive ratio of O(log n) for the k-MPMD problem.
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