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The Maximum Traveling Salesman Problem Under Polyhedral Norms

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Integer Programming and Combinatorial Optimization (IPCO 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1412))

Abstract

We consider the traveling salesman problem when the cities are points in Rd for some fixed d and distances are computed according to a polyhedral norm. We show that for any such norm, the problem of finding a tour of maximum length can be solved in polynomial time. If arithmetic operations are assumed to take unit time, our algorithms run in time O(n f−2 log n), where f is the number of facets of the polyhedron determining the polyhedral norm. Thus for example we have O(n 2 log n) algorithms for the cases of points in the plane under the Rectilinear and Sup norms. This is in contrast to the fact that finding a minimum length tour in each case is NP-hard.

Supported by an Alfred P. Sloan Research Fellowship and NSF grant DMS 9501129.

Supported by the START program Y43-MAT of the Austrian Ministry of Science.

Supported by the NSF through the REU Program.

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Author information

Authors and Affiliations

  1. Dept. of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1009, USA

    Alexander Barvinok & Russell Woodroofe

  2. AT&T Labs — Research, Room C239, Florham Park, NJ, 07932-0971, USA

    David S. Johnson

  3. Institut für Mathematik, TU Graz, Steyrergasse 30, A-8010, Graz, Austria

    Gerhard J. Woeginger

Authors
  1. Alexander Barvinok
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  2. David S. Johnson
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  3. Gerhard J. Woeginger
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  4. Russell Woodroofe
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Editor information

Editors and Affiliations

  1. Department of Computational and Applied Mathematics, Rice University, 6020 Annapolis, Houston, TX, 77005, USA

    Robert E. Bixby

  2. PROS Strategic Solutions, 3223 Smith Street, Houston, TX, 77006, USA

    E. Andrew Boyd

  3. Department of Industrial, Engineering, Texas A&M University, 1000 Country Place Dr. Apt. 69, Houston, TX, 77079, USA

    Roger Z. Ríos-Mercado

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© 1998 Springer-Verlag Berlin Heidelberg

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Barvinok, A., Johnson, D.S., Woeginger, G.J., Woodroofe, R. (1998). The Maximum Traveling Salesman Problem Under Polyhedral Norms. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds) Integer Programming and Combinatorial Optimization. IPCO 1998. Lecture Notes in Computer Science, vol 1412. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-69346-7_15

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  • DOI: https://doi.org/10.1007/3-540-69346-7_15

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64590-0

  • Online ISBN: 978-3-540-69346-8

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