arXiv 2210.031662022-10cs.DS

贪心算法在直线上的在线最小费用匹配中的威力

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

Eric Balkanski, Yuri Faenza, Noemie Perivier

AI总结：

本文研究直线上在线最小费用匹配中贪心算法的性能，证明在随机请求和服务器下其常数竞争，并扩展到服务器过剩和半随机模型，给出紧下界。

摘要：

我们考虑直线上的在线最小费用匹配问题，其中有 $n$ 个服务器，并且在 $n$ 个时间步中的每一步，一个请求到达并且必须不可撤销地匹配到一个尚未匹配的服务器，目标是最小化匹配对之间距离的总和。尽管最坏情况下的竞争比随 $n$ 呈指数增长，但简单的贪心算法（将每个请求匹配到其最近的可用空闲服务器）在实践中表现非常好。因此，一个主要问题是解释贪心算法强大的实证性能。在本文中，我们旨在理解贪心算法在至少部分随机的实例上的性能。当请求和服务器都从 $[0,1]$ 中均匀且独立地抽取时，我们证明贪心算法是常数竞争的，这改进了先前已知的最佳 $O(\sqrt{n})$ 界。我们将这一常数竞争比扩展到具有线性服务器过剩的设置，这改进了先前已知的最佳 $O(\log^3{n})$ 界。此外，我们表明在半随机模型中，请求仍然均匀且独立地抽取，但服务器是敌意选择的，贪心算法实现了 $O(\log{n})$ 的竞争比。当请求以随机顺序到达但被敌意选择时，先前已知贪心算法是 $O(n)$ 竞争的。尽管这种单侧随机性允许贪心算法的竞争比与请求是敌意的且以随机顺序到达的模型相比有大幅改进，但我们通过给出一个紧的 $\Omega(\log{n})$ 下界表明，它不足以获得常数竞争比。这些结果邀请进一步研究在直线上及更广泛范围内，需要多少随机性以及多少随机性足以获得贪心算法在在线最小费用匹配中的强理论保证。

英文摘要：

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 $Ω(\log{n})$ 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.

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