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### PTAS for k-tour cover problem on the plane for moderately large values of k

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Adamaszek, Anna, Czumaj, Artur and Lingas, Andrzej
(2009)
*PTAS for k-tour cover problem on the plane for moderately large values of k.*
In: 20th International Symposium on Algorithms and Computations (ISAAC 2009), Honolulu, HI, December 16-18, 2009. Published in: Lecture Notes in Computer Science, 5878
pp. 994-1003.

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Official URL: http://dx.doi.org/10.1007/978-3-642-10631-6_100

## 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 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 <= 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.

Item Type: | Conference Item (Lecture) |
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Subjects: | Q Science > QA Mathematics Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software |

Divisions: | Faculty of Science > Computer Science |

Library of Congress Subject Headings (LCSH): | Transportation problems (Programming), Approximation algorithms |

Series Name: | Lecture Notes in Computer Science |

Journal or Publication Title: | Lecture Notes in Computer Science |

Publisher: | Springer |

ISBN: | 978-3-642-10630-9 |

ISSN: | 0302-9743 |

Editor: | Dong, Y and Du, DZ and Ibarra, O |

Date: | 2009 |

Volume: | 5878 |

Number of Pages: | 10 |

Page Range: | pp. 994-1003 |

Identification Number: | 10.1007/978-3-642-10631-6_100 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | Engineering and Physical Sciences Research Council (EPSRC), Sweden. Vetenskapsrådet [Research Council], University of Warwick. Centre for Discrete Mathematics and Its Applications |

Grant number: | EP/D063191/1 (EPSRC), 621-2005-408 (VR) |

Version or Related Resource: | Also published in: Adamaszek, A., et al. (2010). PTAS for k-tour cover problem on the plane for moderately large values of k. International Journal of Foundations of Computer Science, 21(6), pp. 893-904. http://wrap.warwick.ac.uk/id/eprint/41591 |

Conference Paper Type: | Lecture |

Title of Event: | 20th International Symposium on Algorithms and Computations (ISAAC 2009) |

Type of Event: | Conference |

Location of Event: | Honolulu, HI |

Date(s) of Event: | December 16-18, 2009 |

Related URLs: | |

References: | 1. Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM 45(5), 753–782 (1998) 2. Asano, T., Katoh, N., Tamaki, H., Tokuyama, T.: Covering points in the plane by k-tours: a polynomial time approximation scheme for fixed k. IBM Tokyo Research Laboratory Research Report RT0162 (1996) 3. Asano, T., Katoh, N., Tamaki, H., Tokuyama, T.: Covering points in the plane by k-tours: Towards a polynomial time approximation scheme for general k. In: Proc. 29th Annual ACM Symposium on Theory of Computing, pp. 275–283 (1997) 4. Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41(1), 153–180 (1994) 5. Dantzig, G.B., Ramser, R.H.: The truck dispatching problem. Management Science 6(1), 80–91 (1959) 6. Das, A.,Mathieu, C.: A quasi-polynomial time approximation scheme for Euclidean capacitated vehicle routing. In: Proc. 21st Annual ACM-SIAM Symposium on Discrete Algorithms (2010) 7. Garey, M.R., Johnson, D.S.: Computers and Intractability. In: A Guide to the Theory of NP-completeness, W.H. Freeman and Company, New York (1979) 8. Haimovich, M., Rinnooy Kan, A.H.G.: Bounds and heuristics for capacitated routing problems. Mathematics of Operation Research 10(4), 527–542 (1985) 9. Laporte, G.: The vehicle routing problem: An overview of exact and approximate algorithms. European Journal of Operational Research 59(3), 345–358 (1992) 10. Mitchell, J.S.B.: Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing 28(4), 1298–1309 (1999) 11. Preparata, F., Shamos, M.: Computational Geometry – an Introduction. Springer, New York (1985) 12. Toth, P., Vigo, D.: The Vehicle Routing Problem. SIAM, Philadelphia (2001) |

URI: | http://wrap.warwick.ac.uk/id/eprint/5481 |

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