Problem
Research questions and friction points this paper is trying to address.
Network Pairing Games
Minimum Cost Perfect Matching
Delay Tolerance
Online matching with delays and stochastic arrival times
🤖 AI Summary
This paper studies the online minimum-cost perfect matching problem under stochastic request arrivals, aiming to minimize the sum of spatial distance costs and temporal delay costs. For request streams modeled as Poisson processes, we propose two deterministic greedy online algorithms—first to break the Ω(log n / log log n) competitive ratio lower bound established for adversarial models, within a stochastic input setting. We prove that both algorithms achieve a constant expected competitive ratio, applicable to any non-decreasing delay function and extendable to variants with rejection penalties. This result demonstrates that stochasticity fundamentally improves matching efficiency, contrasting sharply with polynomial competitive ratios inherent in adversarial settings. It provides the first online policy for delay-sensitive matching systems with provable constant-factor approximation guarantees.
📝 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 $Omega(m^{log frac{3}{2}+varepsilon})$, 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Randomized Minimum Perfect Matching with Delays
Online Algorithms
Stochastic Request Arrival
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