On the Graphical Relaxation of the Symmetric Traveling Salesman (2007)
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BibTeX
@MISC{Oswald07onthe,
author = {Marcus Oswald and Gerhard Reinelt and Dirk Oliver Theis},
title = {On the Graphical Relaxation of the Symmetric Traveling Salesman},
year = {2007}
}
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Abstract
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.
