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On the b-Partite Random Asymmetric Traveling Salesman Problem and Its Assignment Relaxation

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Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques (RANDOM 2001, APPROX 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2129))

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Abstract

We study the relationship between the value of optimal solutions to the random asymmetric b-partite traveling salesman problem and its assignment relaxation. In particular we prove that given a bn×bn weight matrix W = (w ij) such that each finite entry has probability p n of being zero, the optimal values bATSP(W) and AP(W) are equal (almost surely), whenever np n tends to infinity with n. On the other hand, if np n tends to some constant c then ℙ[bATSP(W) ≠ AP(W) > ε > 0, and for np n → 0, ℙ[bATSP(W) ≠ AP(W) → 1 (a.s.). This generalizes results of Frieze, Karp and Reed (1995) for the ordinary asymmetric TSP.

Supported by the graduate school “Effiziente Algorithmen und Mehrskalenmethoden”, Deutsche Forschungsgemeinschaft.

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Authors and Affiliations

  1. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098, Kiel, Germany

    Andreas Baltz, Tomasz Schoen & Anand Srivastav

  2. Adam Mickiewicz University, Poznań, Poland

    Tomasz Schoen

Authors
  1. Andreas Baltz
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  2. Tomasz Schoen
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  3. Anand Srivastav
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, MIT, Cambridge, MA, 02139, USA

    Michel Goemans

  2. Institute for Computer Science and Applied Mathematics, University of Kiel, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Centre Universitaire d’Informatique, Université de Genéve, 24, Rue General Dufour, 1211, Genéve 4, Switzerland

    José D. P. Rolim

  4. Computer Science Division, University of California at Berkeley, 615 Soda Hall, Berkeley, CA, 94720-1776, USA

    Luca Trevisan

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© 2001 Springer-Verlag Berlin Heidelberg

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Baltz, A., Schoen, T., Srivastav, A. (2001). On the b-Partite Random Asymmetric Traveling Salesman Problem and Its Assignment Relaxation. In: Goemans, M., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques. RANDOM APPROX 2001 2001. Lecture Notes in Computer Science, vol 2129. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44666-4_22

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  • DOI: https://doi.org/10.1007/3-540-44666-4_22

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42470-3

  • Online ISBN: 978-3-540-44666-8

  • eBook Packages: Springer Book Archive

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