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On the Steiner 2-edge connected subgraph polytope

Published online by Cambridge University Press:  20 August 2008

A. Rhida Mahjoub  and
Pierre Pesneau
A. Rhida Mahjoub
Affiliation:
LIMOS, CNRS, Univ. Blaise Pascal, Clermont Ferrand II, Complexe Sci. des Cézeaux, 63177 Aubière Cedex; present address: LAMSADE, CNRS, Univ. Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, 75775 Paris Cedex 16, France mahjoub@lamsade.dauphine.fr
Pierre Pesneau
Affiliation:
Université Bordeaux 1, IMB, 351, Cours de la Libération, 33 405 Talence Cedex, France; pierre.pesneau@math.u-bordeaux1.fr
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Abstract

In this paper, we study the Steiner 2-edge connected subgraph polytope. We introduce a large class of valid inequalities for this polytope called the generalized Steiner F-partition inequalities, that generalizes the so-called Steiner F-partition inequalities. We show that these inequalities together with the trivial and the Steiner cut inequalities completely describe the polytope on a class of graphs that generalizes the wheels. We also describe necessary conditions for these inequalities to be facet defining, and as a consequence, we obtain that the separation problem over the Steiner 2-edge connected subgraph polytope for that class of graphs can be solved in polynomial time. Moreover, we discuss that polytope in the graphs that decompose by 3-edge cutsets. And we show that the generalized Steiner F-partition inequalities together with the trivial and the Steiner cut inequalities suffice to describe the polytope in a class of graphs that generalizes the class of Halin graphs when the terminals have a particular disposition. This generalizes a result of Barahona and Mahjoub [4] for Halin graphs. This also yields a polynomial time cutting plane algorithm for the Steiner 2-edge connected subgraph problem in that class of graphs.

Keywords

PolytopeSteiner 2-edge connected graphHalin graph.
Type
Research Article
Information
RAIRO - Operations Research , Volume 42 , Issue 3 , July 2008 , pp. 259 - 283
Copyright
© EDP Sciences, ROADEF, SMAI, 2008

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