This paper presents a new research direction for the Min-cost Perfect Matching with Delays (MPMD) - a problem introduced by Emek et al. (STOC'16). In the original version of this problem, we are given an $n$-point metric space, where requests arrive in an online fashion. The goal is to minimise the matching cost for an even number of requests. However, contrary to traditional online matching problems, a request does not have to be paired immediately at the time of its arrival. Instead, the decision of whether to match a request can be postponed for time $t$ at a delay cost of $t$. For this reason, the goal of the MPMD is to minimise the overall sum of distance and delay costs. Interestingly, for adversarially generated requests, no online algorithm can achieve a competitive ratio better than $O(\log n/\log \log n)$ (Ashlagi et al., APPROX/RANDOM'17). Here, we consider a stochastic version of the MPMD problem where the input requests follow a Poisson arrival process. For such a problem, we show that the above lower bound can be improved by presenting two deterministic online algorithms, which, in expectation, are constant-competitive. The first one is a simple greedy algorithm that matches any two requests once the sum of their delay costs exceeds their connection cost, i.e., the distance between them. The second algorithm builds on the tools used to analyse the first one in order to obtain even better performance guarantees. This result is rather surprising as the greedy approach for the adversarial model achieves a competitive ratio of $\Omega(m^{\log \frac{3}{2}+\varepsilon})$, where $m$ denotes the number of requests served (Azar et al., TOCS'20). Finally, we prove that it is possible to obtain similar results for the general case when the delay cost follows an arbitrary positive and non-decreasing function, as well as for the MPMD variant with penalties to clear pending requests.

翻译：本文提出了关于带延迟的最小成本完美匹配（MPMD）问题的一个新研究方向——该问题由Emek等人（STOC'16）引入。在原始版本中，给定一个$n$点度量空间，请求以在线方式到达，目标是最小化偶数个请求的匹配成本。然而，与传统在线匹配问题不同，请求无需在其到达时立即配对，而是可以将匹配决策延迟时间$t$，并产生延迟成本$t$。因此，MPMD的目标是最小化距离成本与延迟成本的总和。有趣的是，对于对抗性生成的请求，没有任何在线算法能实现优于$O(\log n/\log \log n)$的竞争比（Ashlagi等人，APPROX/RANDOM'17）。本文考虑MPMD问题的随机版本，其中输入请求服从泊松到达过程。针对该问题，我们证明上述下界可通过提出两种确定性在线算法得到改进，这两种算法的期望竞争比为常数。第一种算法是简单的贪婪算法：当任意两个请求的延迟成本之和超过其连接成本（即两者间的距离）时，立即进行匹配。第二种算法基于分析第一种算法时使用的工具，以获得更优的性能保证。这一结果相当令人惊讶，因为在对抗模型下，贪婪方法的竞争比达到$\Omega(m^{\log \frac{3}{2}+\varepsilon})$，其中$m$表示已服务的请求数量（Azar等人，TOCS'20）。最后，我们证明当延迟成本遵循任意正且非递减函数时，以及针对需要清理待处理请求的MPMD变体，有可能获得类似结果。