Generalizations of Matroids

  • Bernhard Korte
  • Jens Vygen
Part of the Algorithms and Combinatorics book series (AC, volume 21)


There are several interesting generalizations of matroids. We have already seen independence systems in Section 13.1, which arose from dropping the axiom (M3). In Section 14.1 we consider greedoids, arising by dropping (M2) instead. Moreover, certain polytopes related to matroids and to submodular functions, called polymatroids, lead to strong generalizations of important theorems; we shall discuss them in Section 14.2. In Sections 14.3 and 14.4 we consider two approaches to the problem of minimizing an arbitrary submodular function: one using the ELLIPSOID METHOD, and one with a combinatorial algorithm. For the important special case of symmetric submodular functions we mention a simpler algorithm in Section 14.5.


Combinatorial Algorithm Submodular Function Incidence Vector Ellipsoid Method Independence System 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany

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