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 polynomial-time 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 <= 2^{{log{\delta}n},} where \delta = \delta(\epsilon). 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/\epsilon)^O(1)) points.