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Circular Ones Matrices and the Stable Set Polytope of Quasi-Line Graphs

  • Conference paper
Book cover Integer Programming and Combinatorial Optimization (IPCO 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3509))

Abstract

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today.

Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. Ben Rebea’s conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that Ben Rebea’s conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph.

In this paper, we give a proof of Ben Rebea’s conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

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Author information

Authors and Affiliations

  1. Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123, Saarbrücken, Germany

    Friedrich Eisenbrand

  2. Dipartimento di Ingegneria dell’Impresa, Universitá di Tor Vergata, via del Politecnico 1, 00165, Roma, Italy

    Gianpaolo Oriolo

  3. Ecole Polytechnique Fédérale de Lausanne, SB-IMA-ROSO Station 8, CH-1015, Lausanne, Switzerland

    Gautier Stauffer

  4. Istituto di Analisi dei Sistemi ed Informatica del CNR, Viale Manzoni 30, 00185, Roma, Italy

    Paolo Ventura

Authors
  1. Friedrich Eisenbrand
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  2. Gianpaolo Oriolo
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  3. Gautier Stauffer
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  4. Paolo Ventura
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Editor information

Editors and Affiliations

  1. Institut für Informatik, Universität zu Köln,  

    Michael Jünger

  2. Fakultät für Mathematik, Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Volker Kaibel

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© 2005 Springer-Verlag Berlin Heidelberg

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Eisenbrand, F., Oriolo, G., Stauffer, G., Ventura, P. (2005). Circular Ones Matrices and the Stable Set Polytope of Quasi-Line Graphs. In: Jünger, M., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2005. Lecture Notes in Computer Science, vol 3509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496915_22

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  • DOI: https://doi.org/10.1007/11496915_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26199-5

  • Online ISBN: 978-3-540-32102-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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