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

  • Anna Adamaszek
  • Artur Czumaj
  • Andrzej Lingas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5878)

Abstract

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 \(\mathcal{NP}\)-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, \(k=\O(\log n / \log\log n)\).

In this paper, we significantly enlarge the set of values of k for which a PTAS is provable. We present a new PTAS for all values of \(k \le 2^{\log^{\delta}n}\), 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 \(\O((k/\epsilon)^{\O(1)})\) points.

Keywords

Vehicle Route Problem Polynomial Time Approximation Scheme Travel Salesperson Problem Small Tour Covering Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Anna Adamaszek
    • 1
  • Artur Czumaj
    • 1
  • Andrzej Lingas
    • 2
  1. 1.Centre for Discrete Mathematics and its Applications (DIMAP) and Department of Computer ScienceUniversity of WarwickUK
  2. 2.Department of Computer ScienceLund UniversityLundSweden

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