Online matching with delays and stochastic arrival times
Mathieu Mari, Michał Pawłowski, Runtian Ren, Piotr Sankowski
TL;DR
This work studies online matching with delays under a stochastic Poisson arrival model for MPMD, showing that simple online strategies can achieve constant expected competitive ratios. It introduces Greedy and Radius algorithms, with ratios $\frac{16}{1-e^{-2}}$ and $\frac{8}{1-e^{-2}}$ respectively, by analyzing the delay cost via a per-point radius $\rho_x$ and relating it to the offline optimum. The radius-based approach anticipates cross-point pairings and extends to general nondecreasing delay functions $f(t)$ and to the MPMD variant with penalties (MPMDfp). Overall, the results demonstrate that stochastic information yields substantially stronger guarantees than adversarial models, with practical implications for real-time matching platforms employing delayed decisions.
Abstract
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 $Ω(m^{\log \frac{3}{2}+\varepsilon})$, 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.