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.