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Probabilistic Analysis of Euclidean Capacitated Vehicle Routing
arXiv - CS - Data Structures and Algorithms Pub Date : 2021-09-14 , DOI: arxiv-2109.06958
Claire Mathieu, Hang Zhou

We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting, where the input distribution consists of $n$ unit-demand customers modeled as independent, identically distributed uniform random points in the two-dimensional plane. The objective is to visit every customer using a set of routes of minimum total length, such that each route visits at most $k$ customers, where $k$ is the capacity of a vehicle. All of the following results are in the random setting and hold asymptotically almost surely. The best known polynomial-time approximation for this problem is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the ITP algorithm is near-optimal when $k$ is either $o(\sqrt{n})$ or $\omega(\sqrt{n})$, and they asked whether the ITP algorithm was also effective in the intermediate range. In this work, we show that when $k=\sqrt{n}$, the ITP algorithm is at best a $(1+c_0)$-approximation for some positive constant $c_0$. On the other hand, the approximation ratio of the ITP algorithm was known to be at most $0.995+\alpha$ due to Bompadre, Dror, and Orlin, where $\alpha$ is the approximation ratio of an algorithm for the traveling salesman problem. In this work, we improve the upper bound on the approximation ratio of the ITP algorithm to $0.915+\alpha$. Our analysis is based on a new lower bound on the optimal cost for the metric capacitated vehicle routing problem, which may be of independent interest.

中文翻译:

欧几里得电容式车辆路由的概率分析

我们对随机设置中的单位需求欧几里得容量车辆路径问题进行概率分析,其中输入分布由 $n$ 个单位需求客户组成,建模为二维平面中独立、同分布的均匀随机点。目标是使用一组最小总长度的路线访问每个客户,使得每条路线最多访问 $k$ 个客户,其中 $k$ 是车辆的容量。以下所有结果都在随机设置中,几乎可以肯定地渐近成立。这个问题最著名的多项式时间近似是迭代旅游分区 (ITP) 算法,由 Haimovich 和 Rinnooy Kan 于 1985 年引入。他们表明,当 $k$ 为 $o(\sqrt {n})$ 或 $\omega(\sqrt{n})$,他们问ITP算法在中间范围内是否也有效。在这项工作中,我们表明,当 $k=\sqrt{n}$ 时,ITP 算法至多是某个正常数 $c_0$ 的 $(1+c_0)$ 近似值。另一方面,由于 Bompadre、Dror 和 Orlin,已知 ITP 算法的逼近比最多为 $0.995+\alpha$,其中 $\alpha$ 是旅行商问题算法的逼近比。在这项工作中,我们将 ITP 算法的近似比的上限提高到 $0.915+\alpha$。我们的分析基于度量容量车辆路由问题的最优成本的新下限,这可能是独立的兴趣。ITP 算法至多是一些正常数 $c_0$ 的 $(1+c_0)$ 近似值。另一方面,由于 Bompadre、Dror 和 Orlin,已知 ITP 算法的逼近比最多为 $0.995+\alpha$,其中 $\alpha$ 是旅行商问题算法的逼近比。在这项工作中,我们将 ITP 算法的近似比的上限提高到 $0.915+\alpha$。我们的分析基于度量容量车辆路由问题的最优成本的新下限,这可能是独立的兴趣。ITP 算法至多是一些正常数 $c_0$ 的 $(1+c_0)$ 近似值。另一方面,由于 Bompadre、Dror 和 Orlin,已知 ITP 算法的逼近比最多为 $0.995+\alpha$,其中 $\alpha$ 是旅行商问题算法的逼近比。在这项工作中,我们将 ITP 算法的近似比的上限提高到 $0.915+\alpha$。我们的分析基于度量容量车辆路由问题的最优成本的新下限,这可能是独立的兴趣。在这项工作中,我们将 ITP 算法的近似比的上限提高到 $0.915+\alpha$。我们的分析基于度量容量车辆路由问题的最优成本的新下限,这可能是独立的兴趣。在这项工作中,我们将 ITP 算法的近似比的上限提高到 $0.915+\alpha$。我们的分析基于度量容量车辆路由问题的最优成本的新下限,这可能是独立的兴趣。
更新日期:2021-09-16
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