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Polynomial-Time Separation of Simple Comb Inequalities

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Integer Programming and Combinatorial Optimization (IPCO 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2337))

Abstract

The comb inequalities are a well-known class of facet-inducing inequalities for the Traveling Salesman Problem, defined in terms of certain vertex sets called the handle and the teeth. We say that a comb inequality is simple if the following holds for each tooth: either the intersection of the tooth with the handle has cardinality one, or the part of the tooth outside the handle has cardinality one, or both. The simple comb inequalities generalize the classical 2-matching inequalities of Edmonds, and also the so-called Chvátal comb inequalities.

In 1982, Padberg and Rao [29] gave a polynomial-time algorithm for separating the 2-matching inequalities — i.e., for testing if a given fractional solution to an LP relaxation violates a 2-matching inequality. We extend this significantly by giving a polynomial-time algorithm for separating the simple comb inequalities. The key is a result due to Caprara and Fischetti.

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Author information

Authors and Affiliations

  1. Department of Management Science, Lancaster University, Lancaster, LA1 4YW, England

    Adam N. Letchford

  2. D.E.I.S., University of Bologna, Viale Risorgimento 2, 40136, Bologna, Italy

    Andrea Lodi

Authors
  1. Adam N. Letchford
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  2. Andrea Lodi
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Editor information

Editors and Affiliations

  1. Program in Applied and Computational Mathematics, University of Princeton, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    William J. Cook

  2. Sloan School of Management, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Andreas S. Schulz

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© 2002 Springer-Verlag Berlin Heidelberg

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Letchford, A.N., Lodi, A. (2002). Polynomial-Time Separation of Simple Comb Inequalities. In: Cook, W.J., Schulz, A.S. (eds) Integer Programming and Combinatorial Optimization. IPCO 2002. Lecture Notes in Computer Science, vol 2337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47867-1_8

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  • DOI: https://doi.org/10.1007/3-540-47867-1_8

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43676-8

  • Online ISBN: 978-3-540-47867-6

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