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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 calledthe generalized Steiner F-partition inequalities, that generalizesthe so-called Steiner F-partition inequalities. We show that theseinequalities together with the trivial and the Steiner cutinequalities completely describe the polytope on a class of graphsthat generalizes the wheels. We also describe necessary conditions forthese inequalities to be facet defining, and as a consequence, weobtain that the separation problem over the Steiner 2-edge connectedsubgraph polytope for that class of graphs can be solved in polynomialtime. Moreover, we discuss that polytope in the graphs that decomposeby 3-edge cutsets. And we show that the generalized SteinerF-partition inequalities together with the trivial and the Steinercut inequalities suffice to describe the polytope in a class of graphsthat generalizes the class of Halin graphs when the terminals have aparticular disposition. This generalizes a result of Barahona andMahjoub [4] for Halin graphs. This also yields apolynomial time cutting plane algorithm for the Steiner 2-edgeconnected subgraph problem in that class of graphs.

Keywords

PolytopeSteiner 2-edge connected graphHalin graph.

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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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