Polynomial time approximation scheme for the capacitated vehicle routing problem with time windows

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 epsilon > 0, finds a (1 + epsilon)-approximate solution for the Eucidean CVRPTW in TIME(TSP, rho, n) + O(n(2)) + O(e(O(q(q/epsilon)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, rho, n) is the complexity of finding a rho-approximation solution of an auxiliary instance of the Euclidean TSP. Thus, the algorithm is a polynomial time approximation scheme for CVRPTW with p(3)q(4) = O(log n) and an efficient polynomial time approximation scheme (EPTAS) for arbitrary fixed values of p and q.