Generalizations of Matroids

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

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.

Keywords

Combinatorial Algorithm Submodular Function Incidence Vector Ellipsoid Method Independence System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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, 1995Google Scholar
  2. Björner, A., and Ziegler, G.M. [1992]: Introduction to greedoids. In: Matroid Applications (N. White, ed.), Cambridge University Press, Cambridge 1992Google Scholar
  3. Frank, A. [2011]: Connections in Combinatorial Optimization. Oxford University Press, Oxford 2011MATHGoogle Scholar
  4. Fujishige, S. [2005]: Submodular Functions and Optimization. Second Edition. Elsevier, Amsterdam 2005MATHGoogle Scholar
  5. Iwata, S. [2008]: Submodular function minimization. Mathematical Programming B 112 (2008), 45–64CrossRefMATHMathSciNetGoogle Scholar
  6. Korte, B., Lovász, L., and Schrader, R. [1991]: Greedoids. Springer, Berlin 1991CrossRefMATHGoogle Scholar
  7. McCormick, S.T. [2004]: Submodular function minimization. In: Discrete Optimization (K. Aardal, G.L. Nemhauser, R. Weismantel, eds.), Elsevier, Amsterdam 2005Google Scholar
  8. Schrijver, A. [2003]: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin 2003, Chapters 44–49Google Scholar
  9. 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. 69–87Google Scholar
  10. 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. 39–49Google Scholar
  11. 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. 185–204Google Scholar
  12. Feige, U., Mirrokni, V.S., and Vondrák, J. [2011]: Maximizing non-monotone submodular functions. SIAM Journal on Computing 40 (2011), 1133–1153CrossRefMATHMathSciNetGoogle Scholar
  13. 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), 107–116Google Scholar
  14. Frank, A. [1981]: A weighted matroid intersection algorithm. Journal of Algorithms 2 (1981), 328–336CrossRefMATHMathSciNetGoogle Scholar
  15. 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. 97–120Google Scholar
  16. 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), 626–628MATHMathSciNetGoogle Scholar
  17. Fujishige, S., Röck, H., and Zimmermann, U. [1989]: A strongly polynomial algorithm for minimum cost submodular flow problems. Mathematics of Operations Research 14 (1989), 60–69CrossRefMATHMathSciNetGoogle Scholar
  18. Grötschel, M., Lovász, L., and Schrijver, A. [1981]: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1 (1981), 169–197CrossRefMATHMathSciNetGoogle Scholar
  19. Grötschel, M., Lovász, L., and Schrijver, A. [1988]: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin 1988MATHGoogle Scholar
  20. Iwata, S. [2002]: A fully combinatorial algorithm for submodular function minimization. Journal of Combinatorial Theory B 84 (2002), 203–212CrossRefMATHMathSciNetGoogle Scholar
  21. Iwata, S. [2003]: A faster scaling algorithm for minimizing submodular functions. SIAM Journal on Computing 32 (2003), 833–840CrossRefMATHMathSciNetGoogle Scholar
  22. Iwata, S., Fleischer, L., and Fujishige, S. [2001]: A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. Journal of the ACM 48 (2001), 761–777CrossRefMATHMathSciNetGoogle Scholar
  23. Jensen, P.M., and Korte, B. [1982]: Complexity of matroid property algorithms. SIAM Journal on Computing 11 (1982), 184–190CrossRefMATHMathSciNetGoogle Scholar
  24. Lovász, L. [1980]: Matroid matching and some applications. Journal of Combinatorial Theory B 28 (1980), 208–236CrossRefMATHGoogle Scholar
  25. 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, 495–517Google Scholar
  26. 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 1983Google Scholar
  27. Nagamochi, H., and Ibaraki, T. [1998]: A note on minimizing submodular functions. Information Processing Letters 67 (1998), 239–244CrossRefMathSciNetGoogle Scholar
  28. Orlin, J.B. [2007]: A faster strongly polynomial time algorithm for submodular function minimization. Mathematical Programming 118 (2009), 237–251CrossRefMATHMathSciNetGoogle Scholar
  29. Queyranne, M. [1998]: Minimizing symmetric submodular functions. Mathematical Programming B 82 (1998), 3–12MATHMathSciNetGoogle Scholar
  30. Rizzi, R. [2000]: On minimizing symmetric set functions. Combinatorica 20 (2000), 445–450CrossRefMATHMathSciNetGoogle Scholar
  31. Schrijver, A. [2000]: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory B 80 (2000), 346–355CrossRefMATHMathSciNetGoogle Scholar
  32. Vygen, J. [2003]: A note on Schrijver’s submodular function minimization algorithm. Journal of Combinatorial Theory B 88 (2003), 399–402CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany

Personalised recommendations