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Separating over classes of TSP inequalities defined by 0 node-lifting in polynomial time

  • Robert Carr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1084)

Abstract

Many important cutting planes nave been discovered for the traveling salesman problem. Until recently (see [5] and [2]), little was known in the way of exact algorithms for separating these inequalities in polynomial time in the size of the fractional point x* which is being separated. Any facet-defining inequality can be neatly categorized by the simple inequality which it is a 0 node-lifting of. Given a class of inequalities consisting of simple inequalities occuring on a fixed sized graph together with all their O node-liftings, an algorithm is presented here that separates over this class of inequalities in polynomial time. This algorithm uses a relaxation of the TSP which we will call cycle-shrink. Cycle-shrink is a compact description of the subtour polytope, and has some other theoretically interesting properties.

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References

  1. 1.
    S. Boyd and W. H. Cunningham (1991), Small travelling salesman polytopes, Mathematics of Operations Research 16 259–271Google Scholar
  2. 2.
    B. Carr, (1995) Separating clique tree and bipartition inequalities having a fixed number of handles and teeth in polynomial time, IPCO proceedings Google Scholar
  3. 3.
    M. Jünger, G. Reinelt, and G. Rinaldi (1994), The traveling salesman problem, Istituto Di Analisi Dei Sistemi Ed Informatica, R. 375, p. 53 Google Scholar
  4. 4.
    M. Jünger, G. Reinelt, and G. Rinaldi (1994), The traveling salesman problem, Istituto Di Analisi Dei Sistemi Ed Informatica, R. 375, p. 59 Google Scholar
  5. 5.
    D. Karger, (1994), Random Sampling in Graph Optimization Problems, PhD Thesis, Department of Computer Science, Stanford University Google Scholar
  6. 6.
    L. Lovasz (1976), On some connectivity properties of Eulerian graphs, Acta Math. Acad. Sci. Hungar., Vol. 28, 129–138CrossRefGoogle Scholar
  7. 7.
    D. Naddef, G. Rinaldi (1992), The graphical relaxation: A new framework for the Symmetric Traveling Salesman Polytope, Mathematical Programming 58, 53–88CrossRefGoogle Scholar
  8. 8.
    M. Yannakakis (1988), Expressing combinatorial optimization problems by linear programs, Proceedings of the 29th IEEE FOCS, 223–228 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Robert Carr
    • 1
  1. 1.Dept. of Computer ScienceUniversity of OttawaOttawaCanada

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