We consider the online minimum cost matching problem on the line, in which 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 to, with the goal of minimizing the sum of the distances between the matched pairs. Despite achieving a worst-case competitive ratio that is exponential in $n$, the simple greedy algorithm, which matches each request to its nearest available free server, performs very well in practice. A major question is thus to explain greedy's strong empirical performance. In this paper, we aim to understand the performance of greedy over instances that are at least partially random. When both the requests and the servers are drawn uniformly and independently from $[0,1]$, we show that greedy is constant competitive, which improves over the previously best-known $O(\sqrt{n})$ bound. We extend this constant competitive ratio to a setting with a linear excess of servers, which improves over the previously best-known $O(\log^3{n})$ bound. 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. When the requests arrive in a random order but are chosen adversarially, it was previously known that greedy is $O(n)$-competitive. Even though this one-sided randomness allows a large improvement in greedy's competitive ratio compared to the model where requests are adversarial and arrive in a random order, we show that it is not sufficient to obtain a constant competitive ratio by giving a tight $\Omega(\log{n})$ 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.

翻译：我们研究在线最小成本直线匹配问题：存在 n 个服务器，在 n 个时间步的每一步中，会到达一个请求，该请求必须被不可撤销地匹配到一个尚未被匹配的服务器，目标是最小化匹配对之间距离的总和。尽管简单贪婪算法（将每个请求匹配到其最近的可用空闲服务器）的最坏情况竞争比是关于 n 的指数级，但在实践中表现非常出色。因此，一个主要问题是解释贪婪算法的强大实证性能。在本文中，我们旨在理解贪婪算法在至少部分随机实例上的性能。当请求和服务器均从 [0,1] 区间独立均匀抽取时，我们证明贪婪算法具有常数竞争比，这改进了先前已知的 O(√n) 上界。我们将此常数竞争比扩展到服务器线性过剩的设置中，改进了先前已知的 O(log³n) 上界。此外，我们证明在半随机模型中（请求仍独立均匀抽取，但服务器由对手选择），贪婪算法实现了 O(log n) 的竞争比。当请求以随机顺序到达但由对手选择时，先前已知贪婪算法是 O(n)-竞争的。尽管与请求由对手选择且以随机顺序到达的模型相比，这种单边随机性使得贪婪算法的竞争比有大幅改进，但我们通过给出一个紧的 Ω(log n) 下界，证明这不足以获得常数竞争比。这些结果促使进一步研究：对于在线最小成本直线匹配（以及更一般情形）中的贪婪算法，需要多少随机性才是必要且充分的，以获得强大的理论保证。