The capacitated vehicle routing problem with time windows (CVRPTW) is a well-known NP-hard combinatorial optimization problem. We present a further development of the approach first proposed by M. Haimovich and A. H. G. Rinnooy Kan and propose an algorithm that, for an arbitrary ε > 0, finds a (1 + ε)-approximate solution for the Euclidean CVRPTW in \(\text{TIME}\;(\text{TSP},\;\rho ,n)\; + \;O({n^2}) + O({e^{(q{{(\tfrac{q}{ \in })}^3}{{(p\rho )}^2}\log (p\rho ))}})\), where q is an upper bound for the capacities of the vehicles, p is the number of time windows, and TIME(TSP, ρ, n) is the complexity of finding a ρ-approximation solution of an auxiliary instance of the Euclidean TSP. Thus, the algorithm is a polynomial-time approximation scheme for the CVRPTW with p³q⁴ = O(log n) and an efficient polynomial-time approximation scheme for arbitrary fixed values of p and q.

中文翻译：

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带时间窗的车辆通行路径问题的多项式时间近似方案

带时间窗的车辆通行能力问题（CVRPTW）是众所周知的NP-hard组合优化问题。我们介绍了M. Haimovich和AHG Rinnooy Kan首先提出的方法的进一步发展，并提出了一种算法，对于任意ε > 0，可以在\（\ text）中找到欧氏CVRPTW的（1 + ε）近似解。{TIME} \;（\ text {TSP}，\; \ rho，n）\; + \; O（{n ^ 2}）+ O（{e ^ {（q {{（\ tfrac {q} { \ in}}} ^ 3} {{（（p \ rho）} ^ 2} \ log（p \ rho）}}））））），其中q是车辆通行能力的上限，p是数量的时间窗口，而TIME（TSP，ρ，n）是找到ρ的复杂度欧氏TSP的辅助实例的近似解。因此，该算法是针对CVRPTW的p ³ q ⁴ = O（log n）的多项式时间近似方案，以及对于p和q的任意固定值的有效多项式时间近似方案。