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On the graphical relaxation of the symmetric traveling salesman polytope

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Abstract

The Graphical Traveling Salesman Polyhedron (GTSP) has been proposed by Naddef and Rinaldi to be viewed as a relaxation of the Symmetric Traveling Salesman Polytope (STSP). It has also been employed by Applegate, Bixby, Chvátal, and Cook for solving the latter to optimality by the branch-and-cut method. There is a close natural connection between the two polyhedra. Until now, it was not known whether there are facets in TT-form of the GTSP polyhedron which are not facets of the STSP polytope as well. In this paper we give an affirmative answer to this question for n ≥ 9. We provide a general method for proving the existence of such facets, at the core of which lies the construction of a continuous curve on a polyhedron. This curve starts in a vertex, walks along edges, and ends in a vertex not adjacent to the starting vertex. Thus there must have been a third vertex on the way.

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Authors and Affiliations

  1. Institute for Computer Science, University of Heidelberg, Im Neuenheimer Feld 368, 69120, Heidelberg, Germany

    Marcus Oswald, Gerhard Reinelt & Dirk Oliver Theis

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  1. Marcus Oswald
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  2. Gerhard Reinelt
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  3. Dirk Oliver Theis
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Correspondence to Marcus Oswald.

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Oswald, M., Reinelt, G. & Theis, D.O. On the graphical relaxation of the symmetric traveling salesman polytope. Math. Program. 110, 175–193 (2007). https://doi.org/10.1007/s10107-006-0060-x

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  • DOI: https://doi.org/10.1007/s10107-006-0060-x

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