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Published in:
2008 | OriginalPaper | Chapter
Generalizations of Matroids
Published in: Combinatorial Optimization
Publisher: Springer Berlin Heidelberg
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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.