Online Min-Cost Matching with General Arrivals

2026-06-03

AI Summary

This work addresses the minimum-cost perfect matching problem under fully online arrivals, where all participants arrive online—a setting that extends beyond the traditional model in which only one side of the market arrives dynamically. Upon arrival, each participant must either be immediately matched to an agent in the waiting pool or join the pool themselves. Under the unknown i.i.d. arrival model, the authors establish for the first time the existence of an online algorithm with a bounded competitive ratio, presenting one that achieves an $O(\log^2 n)$ guarantee. In contrast, they show that under both the random-order and adversarial arrival models, no algorithm can achieve a bounded competitive ratio. This result demonstrates a clear separation in algorithmic performance across different input models, marking a significant theoretical advance in online matching.

Abstract

In the classic online min-cost matching problem, the goal is to match a sequence of requests that arrive dynamically over time to a set of static servers, aiming to minimize the total cost of the matching. This assumes that there are two distinct "sides" and that only one of these sides arrives online, but many of the motivating applications violate these assumptions. We study online min-cost perfect-matching when \emph{all} participants arrive online and, upon arrival, they need to either be matched to someone from a waiting pool or to join the waiting pool. We evaluate the competitive ratios achievable in different input models and show that for both the adversarial and the random-order input models the competitive ratio of any algorithm is unbounded. In contrast, for i.i.d. arrivals we give a $O( \log^2{n})$-competitive algorithm, even if the distribution that generates these arrivals is unknown to the algorithm. This result implies a rare example of separation in the achievable competitive ratio between the random-order and the unknown-i.i.d. input models.

Problem

• online min-cost matching
• perfect matching
• general arrivals
• competitive ratio
• i.i.d. arrivals

Innovation

• online matching
• min-cost perfect matching
• competitive ratio
• i.i.d. arrivals
• unknown distribution