On exchange axioms for valuated matroids and valuated delta-matroids

Abstract

Two further equivalent axioms are given for valuations of a matroid. Let M = (V,B) be a matroid on a finite setV with the family of basesB. For ω:BR the following three conditions are equivalent:

A similar result is obtained for valuations of a delta-matroid.

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References

  1. [1]

    A. Bouchet: Greedy algorithm and symmetric matroids,Mathematical Programming 38 (1987), 147–159.

    Google Scholar 

  2. [2]

    R. Chandrasekaran, andS. N. Kabadi: Pseudomatroids,Discrete Mathematics 71 (1988), 205–217.

    Google Scholar 

  3. [3]

    A. W. M. Dress, andT. Havel: Some combinatorial properties of discriminants in metric vector spaces,Advances in Mathematics 62 (1986), 285–312.

    Google Scholar 

  4. [4]

    A. W. M. Dress, andW. Wenzel: Valuated matroid: A new look at the greedy algorithm,Applied Mathematics Letters 3 (1990), 33–35.

    Google Scholar 

  5. [5]

    A. W. M. Dress, andW. Wenzel: A greedy-algorithm characterization of valuated Δ-matroids,Applied Mathematics Letters 4 (1991), 55–58.

    Google Scholar 

  6. [6]

    A. W. M. Dress, andW. Wenzel: Valuated matroids,Advances in Mathematics 93 (1992), 214–250.

    Google Scholar 

  7. [7]

    K. Murota: Valuated matroid intersection, I: optimality criteria, II: algorithms, to appear inSIAM Journal on Discrete Mathematics,9 (1996), No. 3.

    Google Scholar 

  8. [8]

    K. Murota: Fenchel-type duality for matroid valuations, Report No. 95839-OR, Forschungsinstitut für Diskrete Mathematik, Universität Bonn, 1995.

  9. [9]

    K. Murota: Matroid valuation on independent sets, Report No. 95842-OR, Forschungsinstitut für Diskrete Mathematik, Universität Bonn, 1995.

  10. [10]

    K. Murota: Two algorithms for valuated Δ-matroids,Applied Mathematics Letters,9 (1996) 67–71.

    Google Scholar 

  11. [11]

    K. Murota: Characterizing a valuated delta-matroid as a family of delta-matroids, Report No. 95849-OR, Forschungsinstitut für Diskrete Mathematik, Universität Bonn, 1995.

  12. [12]

    N. Tomizawa: On a self-dual base axiom for matroids (in Japanese); Papers of the Technical Group on Circuit and System Theory, Institute of Electronics and Communication Engineers of Japan, CST77-110 (1977).

  13. [13]

    D. J. A. Welsh:Matroid Theory, Academic Press, London, 1976.

    Google Scholar 

  14. [14]

    W. Wenzel: Pfaffian forms and Δ-matroids,Discrete Mathematics 115 (1993), 253–266.

    Google Scholar 

  15. [15]

    N. White:Theory of Matroids, Cambridge University Press, London, 1986.

    Google Scholar 

Download references

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This work was done while the author was at Forschungsinstitut für Diskrete Mathematik, Universität Bonn.

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Murota, K. On exchange axioms for valuated matroids and valuated delta-matroids. Combinatorica 16, 591–596 (1996). https://doi.org/10.1007/BF01271277

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Mathematics Subject Classification (1991)

  • 05 B 35
  • 90 C 27