The Power of Greedy for Online Minimum Cost Matching on the Line

In the online minimum cost matching problem, there are n servers and, at each of n time steps, a request arrives and must be irrevocably matched to a server that has not yet been matched, with the goal of minimizing the sum of the distances between the matched pairs. Online minimum cost matching is a central problem in applications such as ride-hailing platforms and food delivery services. Despite achieving a worst-case competitive ratio that is exponential in n even on the line, the simple greedy algorithm, which matches each request to its nearest available server, performs well in practice and has a number of attractive features such as strategyproofness. A major question is thus to explain greedy's strong empirical performance. In this paper, we aim to understand the performance of greedy on the line over instances that are at least partially random.

When both the requests and the servers are drawn uniformly and independently from [0, 1], we obtain a constant competitive ratio for greedy, which improves over the previously best-known bound of [EQUATION] for greedy in this setting. We also show that this constant competitive ratio also holds in the excess supply setting where there is a linear excess of servers, which improves over the previously best-known bound of O(log³ n) for greedy in this setting.

We moreover show that in the semi-random model where the requests are still drawn uniformly and independently but where the servers are chosen adversarially, greedy achieves an O(log n) competitive ratio. Even though this one-sided randomness allows a large improvement in greedy's competitive ratio compared to the model where the requests are fully adversarial or arrive in a random order, we show that it is not sufficient to obtain a constant competitive ratio by giving a tight Ω(log n) lower bound. These results invite further investigation about how much randomness is necessary and sufficient to obtain strong theoretical guarantees for the greedy algorithm for online minimum cost matching, on the line and beyond. A full version of this paper can be found at https://arxiv.org/abs/2210.03166.