Skip to main content
SpringerLink
Log in

Generalizations of Matroids

  • Chapter
  • First Online:
Combinatorial Optimization

Part of the book series: Algorithms and Combinatorics ((AC,volume 21))

  • 9732 Accesses

Abstract

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. Finally, in Section 14.6, we discuss the problem of maximizing a submodular function.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

General Literature

  • Bixby , R.E., and Cunningham , W.H. [1995]: Matroid optimization and algorithms. In: Handbook of Combinatorics; Vol. 1 (R.L. Graham , M. Grötschel , L. Lovász , eds.), Elsevier, Amsterdam, 1995

    Google Scholar 

  • Björner , A., and Ziegler , G.M. [1992]: Introduction to greedoids. In: Matroid Applications (N. White , ed.), Cambridge University Press, Cambridge 1992

    Google Scholar 

  • Frank , A. [2011]: Connections in Combinatorial Optimization. Oxford University Press, Oxford 2011

    Google Scholar 

  • Fujishige , S. [2005]: Submodular Functions and Optimization. Second Edition. Elsevier, Amsterdam 2005

    Google Scholar 

  • Iwata , S. [2008]: Submodular function minimization. Mathematical Programming B 112 (2008), 45–64

    Google Scholar 

  • Korte , B., Lovász , L., and Schrader , R. [1991]: Greedoids. Springer, Berlin 1991

    Google Scholar 

  • McCormick , S.T. [2004]: Submodular function minimization. In: Discrete Optimization (K. Aardal , G.L. Nemhauser , R. Weismantel , eds.), Elsevier, Amsterdam 2005

    Google Scholar 

  • Schrijver , A. [2003]: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin 2003, Chapters 44–49

    Google Scholar 

Cited References

  • Buchbinder , N., and Feldman , M. [2016]: Deterministic algorithms for submodular maximization problems. Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (2016), 392–403

    Google Scholar 

  • Buchbinder , N., Feldman , M., Naor , J., and Schwartz , R. [2015]: A tight linear time (1/2)-approximation for unconstrained submodular maximization. SIAM Journal on Computing 44 (2015), 1384–1402

    Google Scholar 

  • Cunningham , W.H. [1984]: Testing membership in matroid polyhedra. Journal of Combinatorial Theory B 36 (1984), 161–188

    Google Scholar 

  • Edmonds , J. [1970]: Submodular functions, matroids and certain polyhedra. In: Combinatorial Structures and Their Applications; Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications 1969 (R. Guy , H. Hanani , N. Sauer , J. Schönheim , eds.), Gordon and Breach, New York 1970, pp. 69–87

    Google Scholar 

  • Edmonds , J. [1979]: Matroid intersection. In: Discrete Optimization I; Annals of Discrete Mathematics 4 (P.L. Hammer , E.L. Johnson , B.H. Korte , eds.), North-Holland, Amsterdam 1979, pp. 39–49

    Google Scholar 

  • Edmonds , J., and Giles , R. [1977]: A min-max relation for submodular functions on graphs. In: Studies in Integer Programming; Annals of Discrete Mathematics 1 (P.L. Hammer , E.L. Johnson , B.H. Korte , G.L. Nemhauser , eds.), North-Holland, Amsterdam 1977, pp. 185–204

    Google Scholar 

  • Feige , U., Mirrokni , V.S., and Vondrák , J. [2011]: Maximizing non-monotone submodular functions. SIAM Journal on Computing 40 (2011), 1133–1153

    Google Scholar 

  • Fleischer , L., and Iwata , S. [2000]: Improved algorithms for submodular function minimization and submodular flow. Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (2000), 107–116

    Google Scholar 

  • Frank , A. [1981]: A weighted matroid intersection algorithm. Journal of Algorithms 2 (1981), 328–336

    Google Scholar 

  • Frank , A. [1982]: An algorithm for submodular functions on graphs. In: Bonn Workshop on Combinatorial Optimization; Annals of Discrete Mathematics 16 (A. Bachem , M. Grötschel , B. Korte , eds.), North-Holland, Amsterdam 1982, pp. 97–120

    Google Scholar 

  • Fujishige , S. [1998]: Another simple proof of the validity of Nagamochi and Ibaraki ’s min-cut algorithm and Queyranne ’s extension to symmetric submodular function minimization. Journal of the Operations Research Society of Japan 41 (1998), 626–628

    Google Scholar 

  • Fujishige , S., Röck , H., and Zimmermann , U. [1989]: A strongly polynomial algorithm for minimum cost submodular flow problems. Mathematics of Operations Research 14 (1989), 60–69

    Google Scholar 

  • Grötschel , M., Lovász , L., and Schrijver , A. [1981]: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1 (1981), 169–197

    Google Scholar 

  • Grötschel , M., Lovász , L., and Schrijver , A. [1988]: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin 1988

    Google Scholar 

  • Iwata , S. [2002]: A fully combinatorial algorithm for submodular function minimization. Journal of Combinatorial Theory B 84 (2002), 203–212

    Google Scholar 

  • Iwata , S. [2003]: A faster scaling algorithm for minimizing submodular functions. SIAM Journal on Computing 32 (2003), 833–840

    Google Scholar 

  • Iwata , S., Fleischer , L., and Fujishige , S. [2001]: A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. Journal of the ACM 48 (2001), 761–777

    Google Scholar 

  • Iwata , S., and Kobayashi , Y. [2017]: A weighted linear matroid parity algorithm. Proceedings of the 49th Annual ACM Symposium on Theory of Computing (2017), 264–276

    Google Scholar 

  • Jensen , P.M., and Korte , B. [1982]: Complexity of matroid property algorithms. SIAM Journal on Computing 11 (1982), 184–190

    Google Scholar 

  • Lee , Y.T., Sidford , A., and Wong , S.C. [2015]: A faster cutting plane method and its implications for combinatorial and convex optimization. Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (2015), 1049–1065

    Google Scholar 

  • Lovász , L. [1980]: Matroid matching and some applications. Journal of Combinatorial Theory B 28 (1980), 208–236

    Google Scholar 

  • Lovász , L. [1981]: The matroid matching problem. In: Algebraic Methods in Graph Theory; Vol. II (L. Lovász , V.T. Sós , eds.), North-Holland, Amsterdam 1981, 495–517

    Google Scholar 

  • Lovász , L. [1983]: Submodular functions and convexity. In: Mathematical Programming: The State of the Art – Bonn 1982 (A. Bachem , M. Grötschel , B. Korte , eds.), Springer, Berlin 1983

    Google Scholar 

  • Nagamochi , H., and Ibaraki , T. [1998]: A note on minimizing submodular functions. Information Processing Letters 67 (1998), 239–244

    Google Scholar 

  • Orlin , J.B. [2007]: A faster strongly polynomial time algorithm for submodular function minimization. Mathematical Programming 118 (2009), 237–251

    Google Scholar 

  • Pap , G. [2017+]. Weighted linear matroid matching. Forthcoming. Abstract in: Oberwolfach Report 53 (2011), 3046–3049

    Google Scholar 

  • Queyranne , M. [1998]: Minimizing symmetric submodular functions. Mathematical Programming B 82 (1998), 3–12

    Google Scholar 

  • Rizzi , R. [2000]: On minimizing symmetric set functions. Combinatorica 20 (2000), 445–450

    Google Scholar 

  • Schrijver , A. [2000]: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory B 80 (2000), 346–355

    Google Scholar 

  • Vygen , J. [2003]: A note on Schrijver ’s submodular function minimization algorithm. Journal of Combinatorial Theory B 88 (2003), 399–402

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute for Discrete Mathematics, University of Bonn, Bonn, Germany

    Bernhard Korte & Jens Vygen

Authors
  1. Bernhard Korte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jens Vygen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer-Verlag GmbH Germany

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Korte, B., Vygen, J. (2018). Generalizations of Matroids. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-56039-6_14

Download citation

Publish with us

Policies and ethics

Search

Navigation