We consider the online Minimum-Cost Perfect Matching with Delays (MPMD) problem introduced by Emek et al. (STOC 2016), in which a general metric space is given, and requests for points in space are submitted in different times in this space by an adversary. The goal is to match requests, while minimizing the sum of distances between matched pairs in addition to the time intervals passed from the moment each request appeared until it is matched. In the online Minimum-Cost Bipartite Perfect Matching with Delays (MBPMD) problem introduced by Ashlagi et al. (APPROX/RANDOM 2017), each request is also associated with one of two classes, and requests can only be matched with requests of the other class. Previous algorithms for the problems mentioned above, include randomized \(O(\log (n))\)-competitive algorithms for known and finite metric spaces, n being the size of the metric space, and a deterministic \(O\left (m\right )\)-competitive algorithm, m being the number of requests. We introduce \(O\left (\frac {1}{\epsilon }m^{\log _{2}\left (\frac {3}{2}+\epsilon \right )}\right )\)-competitive deterministic algorithms for both problems and for any fixed 𝜖 > 0. In particular, for a small enough 𝜖 the competitive ratio becomes \(O\left (m^{0.59}\right )\). These are the first deterministic algorithms for the mentioned online matching problems, achieving a sub-linear competitive ratio. We also show that the analysis of our algorithms is tight. Our algorithms do not need to know the metric space in advance.

中文翻译：

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确定性最小成本匹配延迟

我们考虑了Emek等人提出的在线最小成本时滞完美匹配（MPMD）问题。（STOC 2016），其中给出了一个通用的度量空间，并且对手在该空间中的不同时间提交了对空间点的请求。目标是匹配请求，同时最小化匹配对之间的距离总和，以及从每个请求出现到匹配为止所经过的时间间隔。在Ashlagi等人提出的在线最小成本延迟两部完全匹配（MBPMD）问题中。（APPROX / RANDOM 2017），每个请求也与两个类之一相关联，并且该请求只能与另一个类的请求匹配。解决上述问题的先前算法包括随机\（O（\ log（n））\）-已知和有限度量空间的竞争算法，n是度量空间的大小，以及确定性\（O \ left（m \ right）\）-竞争算法，m是请求数。我们介绍\（O \ left（\ frac {1} {\ epsilon} m ^ {\ log _ {2} \ left（\ frac {3} {2} + \ epsilon \ right}} \ right）\） -针对两个问题以及任何固定的𝜖 > 0的竞争性确定性算法。尤其是对于足够小的𝜖，竞争比变为\（O \ left（m ^ {0.59} \ right）\）。这些是针对上述在线匹配问题的首个确定性算法，可实现亚线性竞争率。我们还表明，对我们算法的分析是严格的。我们的算法不需要事先知道度量空间。