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International Computing and Combinatorics Conference

COCOON 1999: Computing and Combinatorics pp 473–482Cite as

A Fast Approximation Algorithm for TSP with Neighborhoods and Red-Blue Separation

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1627))

Abstract

In TSP with neighborhoods (TSPN) we are given a collection X of k polygonal regions, called neighborhoods, with totally n vertices, and we seek the shortest tour that visits each neighborhood. The Euclidean TSP is a special case of the TSPN problem, so TSPN is also NP-hard. In this paper we present a simple and fast algorithm that, given a start point, computes a TSPN tour of length O(log k) times the optimum in time O(n + k log k). When no start point is given we show how to compute a “good” start point in time O(n 2 log n), hence we obtain a logarithmic approximation algorithm that runs in time O(n 2 log n). We also present an algorithm which performs at least one of the following two tasks (which of these tasks is performed depends on the given input): (1) It outputs in time O(n log n) a TSPN tour of length O(log k) times the optimum. (2) It outputs a TSPN tour of length less than (1+∈) times the optimum in cubic time, where ∈ is an arbitrary real constant given as an optional parameter.

The results above are significant improvements, since the best previously known logarithmic approximation algorithm runs in Ω(n 5) time in the worst case.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Lund University, Box 118, S-221 00, Lund, Sweden

    Joachim Gudmundsson & Christos Levcopoulos

Authors
  1. Joachim Gudmundsson
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  2. Christos Levcopoulos
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Editor information

Editors and Affiliations

  1. Department of Information and System Engineering Faculty of Science and Engineering, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan

    Takano Asano

  2. Department of Information Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

    Hideki Imai

  3. Academia Sinica, Institute of Information Science, Nankang, Taipei, Taiwan

    D. T. Lee

  4. Department of Computer Science Faculty of Engineering, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma, 376-8515, Japan

    Shin-ichi Nakano

  5. IBM Tokyo Research Laboratory, 1623-14, Shimo-Tsuruma, Yamato Kanagawa, 242-0001, Japan

    Takeshi Tokuyama

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© 1999 Springer-Verlag Berlin Heidelberg

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Gudmundsson, J., Levcopoulos, C. (1999). A Fast Approximation Algorithm for TSP with Neighborhoods and Red-Blue Separation. In: Asano, T., Imai, H., Lee, D.T., Nakano, Si., Tokuyama, T. (eds) Computing and Combinatorics. COCOON 1999. Lecture Notes in Computer Science, vol 1627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48686-0_47

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  • DOI: https://doi.org/10.1007/3-540-48686-0_47

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66200-6

  • Online ISBN: 978-3-540-48686-2

  • eBook Packages: Springer Book Archive

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