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One-third-integrality in the max-cut problem

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Abstract

Given a graphG = (V, E), the metric polytopeS (G) is defined by the inequalitiesx(F) − x(C∖F) ⩽ |F| − 1 for\(F \subseteq C\), |F| odd,C cycle ofG, and 0 ⩽x e ⩽ 1 fore ∈ E. Optimization overS (G) provides an approximation for the max-cut problem. The graphG is called 1/d-integral if all the vertices ofS(G) have their coordinates in{i/d ∣ 0 ⩽ i ⩽ d}. We prove that the class of 1/d-integral graphs is closed under minors, and we present several minimal forbidden minors for 1/3-integrality. In particular, we characterize the 1/3-integral graphs on seven nodes. We study several operations preserving 1/d-integrality, in particular, thek-sum operation for 0 ⩽k ⩽ 3. We prove that series parallel graphs are characterized by the following stronger property. All vertices of the polytopeS (G) ∩ {x ∣ ℓ ⩽ x ⩽ u} are 1/3-integral for every choice of 1/3-integral boundsℓ, u on the edges ofG.

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Authors and Affiliations

  1. Département de Mathématiques et d'Informatique, Ecole Normale Supérieure, CNRS, LIENS, 45 rue d'Ulm, 75230, Paris cedex 05, France

    Monique Laurent

  2. Department für Informatik, Universität Passau, Passau, Germany

    Svatopluk Poljak

Authors
  1. Monique Laurent
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  2. Svatopluk Poljak
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Additional information

Research by this author was partially done at CWI in Amsterdam.

Research by this author was done at the Institut für Diskrete Mathematik of Bonn, supported by the A. von Humboldt Foundation.

Deceased on April 2nd, 1995.

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Laurent, M., Poljak, S. One-third-integrality in the max-cut problem. Mathematical Programming 71, 29–50 (1995). https://doi.org/10.1007/BF01592243

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