Abstract

The capacitated vehicle routing problem (CVRP) is a classical combinatorial optimization problem having a wide range of practically important applications in operations research. As most combinatorial problems, CVRP is strongly NP-hard (even on the Euclidean plane). A metric instance of CVRP is APX-complete, so it cannot be approximated to arbitrary prescribed accuracy in the class of polynomial time algorithms (assuming that \(P \ne NP\)). Nevertheless, in the case of finite-dimensional Euclidean spaces, a quasi-polynomial or even polynomial time approximation scheme can be found for the problem by applying an approach based on works by S. Arora, A. Das, and C. Mathieu. Below, this approach has been extended for the first time to a significantly larger class of metric spaces of fixed doubling dimension. It is shown that CVRP formulated in such a space has a quasi-polynomial time approximation scheme whenever the number of routes in its optimal solution is bounded above by a polynomial in the logarithm of the input size.

中文翻译：

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在固定加倍维度的度量空间中有限路径数量的有容量车辆路径问题的逼近

摘要

有能力的车辆路径问题（CVRP）是一个经典的组合优化问题，在运筹学中具有广泛的实际重要应用。与大多数组合问题一样，CVRP 是强 NP 难的（即使在欧几里得平面上）。CVRP 的度量实例是 APX 完全的，因此它不能近似到多项式时间算法类中的任意规定精度（假设\(P\ne NP\)）。然而，在有限维欧几里得空间的情况下，可以通过应用基于 S. Arora、A. Das 和 C. Mathieu 作品的方法，为问题找到拟多项式甚至多项式时间近似方案。下面，这种方法第一次被扩展到一个明显更大的一类固定倍增维的度量空间。结果表明，只要其最优解中的路径数以输入大小的对数形式的多项式为界，则在这样的空间中制定的 CVRP 具有准多项式时间近似方案。