PTAS for k-Tour Cover Problem on the Plane for Moderately Large Values of k

Let P be a set of n points in the Euclidean plane and let O be the origin point in the plane. In the k-tour cover problem (called frequently the capacitated vehicle routing problem), the goal is to minimize the total length of tours that cover all points in P , such that each tour starts and ends in O and covers at most k points from P . The k-tour cover problem is known to be NP-hard. It is also known to admit constant factor approximation algorithms for all values of k and even a polynomialtime approximation scheme (PTAS) for small values of k, i.e., k = O(log n/ log log n). We significantly enlarge the set of values of k for which a PTAS is provable. We present a new PTAS for all values of k ≤ 2logδ n, where δ = δ(ε). The main technical result proved in the paper is a novel reduction of the k-tour cover problem with a set of n points to a small set of instances of the problem, each with O((k/ε)O(1)) points.