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A general model for matroids and the greedy algorithm

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Abstract

We present a general model for set systems to be independence families with respect to set families which determine classes of proper weight functions on a ground set. Within this model, matroids arise from a natural subclass and can be characterized by the optimality of the greedy algorithm. This model includes and extends many of the models for generalized matroid-type greedy algorithms proposed in the literature and, in particular, integral polymatroids. We discuss the relationship between these general matroids and classical matroids and provide a Dilworth embedding that allows us to represent matroids with underlying partial order structures within classical matroids. Whether a similar representation is possible for matroids on convex geometries is an open question.

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Authors and Affiliations

  1. Mathematisches Institut/ZAIK, Universität zu Köln, 50931, Cologne, Germany

    Ulrich Faigle

  2. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Satoru Fujishige

Authors
  1. Ulrich Faigle
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  2. Satoru Fujishige
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Corresponding author

Correspondence to Ulrich Faigle.

Additional information

S. Fujishige’s research was supported by a Grant-in-Aid for Scientific Research of the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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Faigle, U., Fujishige, S. A general model for matroids and the greedy algorithm. Math. Program. 119, 353–369 (2009). https://doi.org/10.1007/s10107-008-0213-1

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  • DOI: https://doi.org/10.1007/s10107-008-0213-1

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