Abstract
Many combinatorial optimization problems can be formulated as follows. Given a set system (E, ℱ), i. e. a finite set E and some ℱ ⊆ 2E, and a cost function c: ℱ → ℝ, find an element of ℱ whose cost is minimum or maximum.
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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
Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., and Schrijver, A. [1998]: Combinatorial Optimization. Wiley, New York 1998, Chapter 8
Faigle, U. [1987]: Matroids in combinatorial optimization. In: Combinatorial Geometries (N. White, ed.), Cambridge University Press, 1987
Gondran, M., and Minoux, M. [1984]: Graphs and Algorithms. Wiley, Chichester 1984, Chapter 9
Lawler, E.L. [1976]: Combinatorial Optimization; Networks and Matroids. Holt, Rinehart and Winston, New York 1976, Chapters 7 and 8
Oxley, J.G. [1992]: Matroid Theory. Oxford University Press, Oxford 1992
von Randow, R. [1975]: Introduction to the Theory of Matroids. Springer, Berlin 1975
Recski, A. [1989]: Matroid Theory and its Applications. Springer, Berlin, 1989
Welsh, D.J.A. [1976]: Matroid Theory. Academic Press, London 1976
Cited References
Cunningham, W.H. [1986]: Improved bounds for matroid partition and intersection algorithms. SIAM Journal on Computing 15 (1986), 948–957
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. Schonheim, eds.), Gordon and Breach, New York 1970, pp. 69–87
Edmonds, J. [1971]: Matroids and the greedy algorithm. Mathematical Programming 1 (1971), 127–136
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
Edmonds, J., and Fulkerson, D.R. [1965]: Transversals and matroid partition. Journal of Research of the National Bureau of Standards B 69 (1965), 67–72
Frank, A. [1981]: A weighted matroid intersection algorithm. Journal of Algorithms 2 (1981), 328–336
Frank, A., and Tardos, É. [1989]: An application of submodular flows. Linear Algebra and Its Applications 114/115 (1989), 329–348
Fulkerson, D.R. [1971]: Blocking and anti-blocking pairs of polyhedra. Mathematical Programming 1 (1971), 168–194
Gabow, H.N. [1991]: A matroid approach to finding edge connectivity and packing arborescences. Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science (1991), 112-122
Gabow, H.N., and Xu, Y. [1996]: Efficient theoretic and practical algorithms for linear matroid intersection problems. Journal of Computer and System Sciences 53 (1996), 129–147
Hausmann, D., Jenkyns, T.A., and Korte, B. [1980]: Worst case analysis of greedy type algorithms for independence systems. Mathematical Programming Study 12 (1980), 120–131
Hausmann, D., and Korte, B. [1981]: Algorithmic versus axiomatic definitions of matroids. Mathematical Programming Study 14 (1981), 98–111
Jenkyns, T.A. [1976]: The efficiency of the greedy algorithm. Proceedings of the 7th SE Conference on Combinatorics, Graph Theory, and Computing, Utilitas Mathematica, Winnipeg 1976, pp. 341–350
Korte, B., and Hausmann, D. [1978]: An analysis of the greedy algorithm for independence systems. In: Algorithmic Aspects of Combinatorics; Annals of Discrete Mathematics 2 (B. Alspach, P. Hell, D.J. Miller, eds.), North-Holland, Amsterdam 1978, pp. 65–74
Korte, B., and Monma, C.L. [1979]: Some remarks on a classification of oracle-type algorithms. In: Numerische Methoden bei graphentheoretischen und kombinatorischen Problemen; Band 2 (L. Collatz, G. Meinardus, W. Wetterling, eds.), Birkhäuser, Basel 1979, pp. 195–215
Lehman, A. [1979]: On the width-length inequality. Mathematical Programming 17 (1979), 403–417
Nash-Williams, C.S.J.A. [1967]: An application of matroids to graph theory. In: Theory of Graphs; Proceedings of an International Symposium in Rome 1966 (P. Rosenstiehl, ed.), Gordon and Breach, New York, 1967, pp. 263–265
Rado, R. [1942]: A theorem on independence relations. Quarterly Journal of Math. Oxford 13 (1942), 83–89
Rado, R. [1957]: Note on independence functions. Proceedings of the London Mathematical Society 7 (1957), 300–320
Seymour, P.D. [1977]: The matroids with the Max-Flow Min-Cut property. Journal of Combinatorial Theory B 23 (1977), 189–222
Whitney, H. [1933]: Planar graphs. Fundamenta Mathematicae 21 (1933), 73–84
Whitney, H. [1935]: On the abstract properties of linear dependence. American Journal of Mathematics 57 (1935), 509–533
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Korte, B., Vygen, J. (2002). Matroids. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21711-5_13
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DOI: https://doi.org/10.1007/978-3-662-21711-5_13
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