Polynomial-Time Approximation Scheme for the Capacitated Vehicle Routing Problem with Time Windows

Abstract

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.