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Polynomial-Time Approximation Scheme for the Capacitated Vehicle Routing Problem with Time Windows
Proceedings of the Steklov Institute of Mathematics ( IF 0.4 ) Pub Date : 2020-03-22 , DOI: 10.1134/s0081543819070058
M. Yu. Khachai , Yu. Yu. Ogorodnikov

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:

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,ρn)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pq的任意固定值的有效多项式时间近似方案。
更新日期:2020-03-22
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