Deterministic Primal-Dual Algorithms for Online k-Way Matching with Delays

In this paper, we study the Min-cost Perfect k-way Matching with Delays (k-MPMD), recently introduced by Melnyk et al. In the problem, m requests arrive one-by-one over time in a metric space. At any time, we can irrevocably make a group of k requests who arrived so far, that incurs the distance cost among the k requests in addition to the sum of the waiting cost for the k requests. The goal is to partition all the requests into groups of k requests, minimizing the total cost. The problem is a generalization of the min-cost perfect matching with delays (corresponding to 2-MPMD). It is known that no online algorithm for k-MPMD can achieve a bounded competitive ratio in general, where the competitive ratio is the worst-case ratio between its performance and the offline optimal value. On the other hand, k-MPMD is known to admit a randomized online algorithm with competitive ratio \(O(k^{5}\log n)\) for a certain class of k-point metrics called the H-metric, where n is the size of the metric space. In this paper, we propose a deterministic online algorithm with a competitive ratio of \(O(mk^2)\) for the k-MPMD in H-metric space. Furthermore, we show that the competitive ratio can be improved to \(O(m + k^2)\) if the metric is given as a diameter on a line.