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Verallgemeinerungen von Matroiden

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Kombinatorische Optimierung

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Zusammenfassung

Es gibt einige interessanteVerallgemeinerungen von Matroiden. Unabhängigkeitssysteme haben wir in Abschnitt 13.1 bereits kennen gelernt; sie entstehen durch Weglassen des Axioms (M3). In Abschnitt 14.1 betrachten wir Greedoide; sie entstehen durch Weglassen des Axioms (M2) (anstatt von (M3)). Ferner gibt es gewisse mit Matroiden und submodularen Funktionen verwandte Polytope - so genannte Polymatroide - die zu starken Verallgemeinerungen von wichtigen Sätzen führen; diese werden wir in Abschnitt 14.2 betrachten. In den Abschnitten 14.3 und 14.4 werden wir zwei verschiedene Zugänge zu dem Problem der Minimierung einer beliebigen submodularen Funktion kennen lernen: einen über die ELLIPSOIDMETHODE und einen anderen mittels eines kombinatorischen Algorithmus. Für den wichtigen Spezialfall einer symmetrischen submodularen Funktion werden wir in Abschnitt 14.5 einen einfacheren Algorithmus angeben.

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Author notes

  1. Bernhard Korte & Jens Vygen

    Present address: Forschungsinstitut für Diskrete Mathematik, Universität Bonn, Lennestr. 2, 53113, Bonn, Deutschland

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    1. Bernhard Korte
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    2. Jens Vygen
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    Correspondence to Bernhard Korte or Jens Vygen .

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    Korte, B., Vygen, J. (2012). Verallgemeinerungen von Matroiden. In: Kombinatorische Optimierung. Springer-Lehrbuch Masterclass. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25401-7_14

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