Skip to main content
SpringerLink
Log in

Applications of Matroid Theory

  • Chapter
Mathematical Programming The State of the Art

Abstract

It will be shown that looking at a problem from the viewpoint of matroids enables us to understand the essence of the problem as well as its relations to other problems, clearly, preventing us from probable confusion into which we might have been involved without matroids, and that mathematical techniques developed in matroid theory are powerful for manipulating and solving the mathematical model which would otherwise have been impossible, or at best prohibitively complicated. Examples of problems to be discussed:

  1. 1.

    Topological, geometrical and physical matroids, or faithful and unfaithful representations in terms of matroids

  2. 2.

    Elements and their interconnections

  3. 3.

    Minimum-size systems of equations

  4. 4.

    Structural solvability of systems of equations

  5. 5.

    Two kinds of dualities

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 PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  • Aigner, M. (1979): Combinatorial Theory. Springer-Verlag.

    Google Scholar 

  • Amari, S. (1962): Topological foundations of Kron’s tearing of electric networks; Information-theoretical foundations of diakoptics and codiakoptics. RAAG Memoirs, Vol. 3, F-VI; F-VII, pp. 322–350; 351–371.

    Google Scholar 

  • Asimov, L., and Roth, B. (1978): Rigidity of graphs. Transactions of the American Mathematical Society, Vol. 245, pp. 279–289.

    Article  MathSciNet  Google Scholar 

  • Asimov, L., and Roth, B. (1979): Rigidity of graphs, II. Journal of Mathematical Analysis and Applications, Vol. 68, pp. 171–190.

    Article  MathSciNet  Google Scholar 

  • Birkhoff, G. (1935): Sur les espaces discrets. Computes Rendus de l’Academie des Sciences, Paris, Tome 201, pp. 19–20.

    Google Scholar 

  • Birkhoff, G. (1967): Lattice Theory ( third edition ). American Mathematical Society Colloquium Publications, Providence.

    Google Scholar 

  • Bixby, R. E. (1981): Hidden structure in linear programs. In Computer-Assisted Analysis and Model Simplification (Greenburg, H. J., and Maybee, J. S., eds. ), Academic Press, pp. 327–360.

    Google Scholar 

  • Bolker, E. D., and Crapo, H. (1977): How to brace a one-story-building. Environment and Planing B, Vol. 4, pp. 125–152.

    Google Scholar 

  • Bruno, J., and Weinberg, L. (1970): A constructive graph-theoretic solution of the Shannon switching game. IEEE Transactions on Circuit Theory, Vol. CT-17, pp. 74–81.

    Google Scholar 

  • Bruno, J., and Weinberg, L. (1971): The principal minors of a matroid. Linear Algebra and Its Applications, Vol. 4, pp. 17–54.

    Article  MathSciNet  MATH  Google Scholar 

  • Bruno, J., and Weinberg, L. (1976): Generalized networks: networks embedded on a matroid, I and II. Networks, Vol. 6, pp. 53–94, pp. 231–272.

    Google Scholar 

  • Bruter, C. P. (1974): Eléments de la Théorie des Matroïdes. Lecture Notes in Mathematics 387, Springer-Verlag, Berlin.

    Google Scholar 

  • Bryant, P. R. (1962): The explicit form of Bashkow’s A-matrix. IRE Transactions on Circuit Theory, Vol. CT-9, pp. 303 - 306.

    Google Scholar 

  • Camerini, P. M., Fratta, L., and Maffioli, F. (1979): A note on finding optimum branchings. Networks, Vol. 9, pp. 309–312.

    Article  MathSciNet  MATH  Google Scholar 

  • Chu, Y.-J., and Liu, T.-H. (1965): On the shortest arborescence of a directed graph. Scientia Sinica, Vol. 14, pp. 1396–1400.

    MathSciNet  MATH  Google Scholar 

  • Crapo, H. (1979): Structural rigidity. Structural Topology, Vol. 1, pp. 26–45.

    MathSciNet  MATH  Google Scholar 

  • Crapo, H., and Rota, G.-C. (1971): Combinatorial Geometries. M.I.T. Press.

    Google Scholar 

  • Csurgay, A., Kovács, Z., and Recski, A. (1974): Transient analysis of lumped-distributed nonlinear networks. Proceedings of the 5 th International Colloquium on Microwave Communication, Budapest.

    Google Scholar 

  • Dulmage, A. L., and Mendelsohn, N. S. (1958): Coverings of bipartite graphs. Canadian Journal of Mathematics, Vol. 10, pp. 517–534.

    Article  MathSciNet  MATH  Google Scholar 

  • Dulmage, A. L., and Mendelsohn, N. S. (1959): A structure theory of bipartite graphs of finite exterior dimension. Transactions of the Royal Society of Canada, Third Series, Section III, Vol. 53, pp. 1–13.

    Google Scholar 

  • Dulmage, A. L., and Mendelsohn, N. S. (1962): On the inversion of sparse matrices. Mathematics of Computation, Vol. 16, pp. 494 - 496.

    Article  MathSciNet  MATH  Google Scholar 

  • Dulmage, A. L., and Mendelsohn, N. S. (1963): Two algorithms for bipartite graphs. Journal of the Society of the Industrial and Applied Mathematics, Vol. 11, pp. 183– 194.

    Google Scholar 

  • Edmonds, J. (1965): Minimum partition of a matroid into independent subsets. Journal of Research of the National Bureau of Standards, Vol. 69 B, pp. 73 - 77.

    Google Scholar 

  • Edmonds, J. (1967): Optimum branchings. Journal of Research of the National Bureau of Standards, Vol 71B, pp. 233 - 240.

    MathSciNet  MATH  Google Scholar 

  • Edmonds, J. (1970): Submodular functions, matroids and certain polyhedra. Proceedings of the International Conference on Combinatorial Structures and Their Applications (Guy, R., et al. eds.), Gordon and Breach, New York, pp. 69–87.

    Google Scholar 

  • Edmonds, J. (1971): Matroids and the greedy algorithm. Mathematical Programming. Vol. 1, pp. 127 - 136.

    Article  MathSciNet  MATH  Google Scholar 

  • Ford, L. R., Jr., and Fulkerson, D. R. (1962): Flows in Networks. Princeton University Press, Princeton, New Jersey.

    Google Scholar 

  • Fujishige, S. (1977 a): A primal approach to the independent assignment problem. Journal of the Operations Research Society of Japan, Vol. 20, pp. 1–15.

    Google Scholar 

  • Fujishige, S. (1977 b): An algorithm for finding an optimal independent linkage. Journal of the Operations Research Society of Japan, Vol. 20, pp. 59–75.

    Google Scholar 

  • Fujishige, S. (1978 a): Algorithms for solving the independent-flow problems. Journal of the Operations Research Society of Japan, Vol. 21, pp. 189–203.

    Google Scholar 

  • Fujishige, S. (1978 b): Polymatroidal dependence structure of a set of random variables. Information and Control, Vol. 39, pp. 55–72.

    Google Scholar 

  • Fujishige, S. (1980 a): Principal structures of submodular systems. Discrete Applied Mathematics, Vol. 2, pp. 77 - 79.

    Google Scholar 

  • Fujishige, S. (1980 b): An efficient PQ-graph algorithm for solving the graph-realization problem. Journal of Computer and System Sciences, Vol. 21, pp. 63–86.

    Google Scholar 

  • Fujishige, S. (1980 c): Lexicographically optimal base of a polymatroid with respect to a weight vector. Mathematics of Operations Research, Vol. 5, pp. 186–196.

    Google Scholar 

  • Fujushige, S., Ohkubo, K., and Iri, M. (1978): Optimal flows in a network with several sources and sinks (in Japanese). Proceedings of the 1978 Fall Conference of the Operations Research Society of Japan, A-2, pp. 22–23.

    Google Scholar 

  • Fulkerson, D. R. (1971): Blocking and anti-blocking pairs of polyhedra. Mathematical Programming, Vol. 1, pp. 168–194.

    Article  MathSciNet  MATH  Google Scholar 

  • Han, T.-S. (1979 a): The capacity region of general multi-access channel with certain correlated sources. Information and Control, Vol. 40, pp. 37–60.

    Google Scholar 

  • Han, T.-S. (1979 b): Source coding with cross observation at the encoders. IEEE Transactions on Information Theory, Vol. IT-25, pp. 360–361.

    Google Scholar 

  • Han, T.-S. (1980): Slepian-Wolf-Cover theorem for networks of channels. Information and Control, Vol. 47, pp. 67–83.

    Article  MathSciNet  MATH  Google Scholar 

  • Harary, F., and Welsh, D. (1969): Matroids versus graphs. The Many Facets of Graph Theory, Springer Lecture Notes 110, p. 115.

    Google Scholar 

  • Hausmann, D., and Korte, B. (1978): Lower bounds on the worst-case complexity of some oracle algorithms. Discrete Mathematics, Vol. 24, pp. 261–276.

    Article  MathSciNet  MATH  Google Scholar 

  • Holzmann, C. A. (1977): Realization of netoids. Proceedings of the 20 th Midwest Symposium on Circuits and Systems, Lubbock, Texas, pp. 394–398.

    Google Scholar 

  • Holzmann, C. A. (1979): Binary netoids. Proceedings of the International Symposium on Circuits and Systems, Tokyo, pp. 1000–1003.

    Google Scholar 

  • Imai, H. (1983): Efficient Solutions for Combinatorial Optimization Problems by Means of Network-Flow Algorithms (in Japanese). Master’s Thesis, Department of Mathematical Engineering and Instrumentation Physics, University of Tokyo.

    Google Scholar 

  • Iri, M. (1962): A necessary and sufficient condition for a matrix to be the loop or cut-set matrix of a graph and a practical method for the topological synthesis of networks. RAAG Research Notes, Third Series, No. 50.

    Google Scholar 

  • Iri, M. (1966): A criterion for the reducibility of a linear programming problem to a linear network-flow problem. RAAG Research Notes, Third Series, No. 98.

    Google Scholar 

  • Iri, M. (1968 a): On the synthesis of loop and cutset matrices and the related problems. RAAG Memoirs, Vol. 4, A-XIII, pp. 4-38.

    Google Scholar 

  • Iri, M. (1968b): A min-max theorem for the ranks and term-ranks of a class of matrices: An algebraic approach to the problem of the topological degrees of freedom of a network (in Japanese). Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. 51 A, pp. 180 - 187.

    Google Scholar 

  • Iri, M. ( 1968 c): A critical review of the matroid-theoretical and the algebraic-topological theory of networks. RAAG Memoirs, Vol. IV, Division A, pp. 39 - 46.

    Google Scholar 

  • Iri, M. ( 1968 d): Metatheoretical considerations on duality. RAAG Research Notes, Third Series, No. 124.

    Google Scholar 

  • Iri, M. ( 1969 a): Network Flow, Transportation and Scheduling - Theory and Algorithms. Academic Press, New York.

    Google Scholar 

  • Iri, M. (1969 b): The maximum-rank minimum-term-rank theorem for the pivotal transforms of a matrix. Linear Algebra and Its Applications, Vol. 2, pp. 427–446.

    Google Scholar 

  • Iri, M. (1971): Combinatorial canonical form of a matrix with applications to the principal partition of a graph (in Japanese). Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. 54 A, pp. 30–37.

    Google Scholar 

  • Iri, M. (1978): A practical algorithm for the Menger-type generalization of the independent assignment problem. Mathematical Programming Study, Vol. 8, pp. 88–105.

    MathSciNet  Google Scholar 

  • Iri, M. (1979 a): Survey of recent trends in applications of matroids. Proceedings of the 1979 International Symposium on Circuits and Systems, Tokyo, p. 987.

    Google Scholar 

  • Iri, M. (1979 b): A review of recent work in Japan on principal partitions of matroids and their applications. Annals of the New York Academy of Sciences, Vol. 319 (Proceedings of the Second International Conference on Combinatorial Mathematics, 1978 ), pp. 306 - 319.

    Google Scholar 

  • Iri, M. (1981a): “Dualities” in graph theory and in the related fields viewed from the metatheoretical standpoint. In Graph Theory and Algorithms (Saito, N., and Nishizeki, T., eds.). Lecture Note in Computer Science 108, Springer-Verlag, pp. 124–136.

    Google Scholar 

  • Iri, M. (1981b): Application of matroid theory to engineering systems problems. Proceedings of the Sixth Conference on Probability Theory (September, 1979; Bereanu, B., et al., eds.), Editura Academiei Republicii Socialiste Romania, pp. 107–127.

    Google Scholar 

  • Iri, M. (1982): Structural theory for the combinatorial systems characterized by sub-modular functions. Silver-Jubilee Conference on Combinatorics, The University of Waterloo, June.

    Google Scholar 

  • Iri, M., and Fujishige, S. (1981): Use of matroid theory in operations research, circuits and systems theory. International Journal of Systems Science, Vol. 12, pp. 27–54.

    Article  MathSciNet  MATH  Google Scholar 

  • Iri, M., and Han, T.-S. (1977): Linear Algebra - Standard Forms of Matrices (in Japanese). Kyoiku-Shuppan Co., Tokyo.

    Google Scholar 

  • Iri, M., and Recski, A. (1979): Reflection on the concepts of dual, inverse and adjoint networks (in Japanese). Papers of the Technical Group on Circuits and Systems, Institute of Electronics and Communication Engineers of Japan, CAS 79-78 (English translation available).

    Google Scholar 

  • Iri, M., and Recski, A. (1980 a): Reflection on the concepts of dual, inverse and adjoint networks, II - Towards a qualitative theory (in Japanese). Papers of the Technical Group on Circuits and Systems, Institute of Electronics and Communication Engineers of Japan, CAS 79-133 (English translation available).

    Google Scholar 

  • Iri, M., and Recski, A. (1980b): What does duality really mean? Circuit Theory and Applications, Vol. 8, pp. 317–324.

    Article  MathSciNet  MATH  Google Scholar 

  • Iri, M., and Tomizawa, N. (1974 a): A practical criterion for the existence of the unique solution in a linear electric network with mutual couplings (in Japanese). Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. 57 A, pp. 599–605.

    Google Scholar 

  • Iri, M., and Tomizawa, N. (1974b): An algorithms for solving the ‘independent assignment problem’ with application to the problem of determining the order of complexity of a network (in Japanese). Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. 57 A, pp. 627–629.

    Google Scholar 

  • Iri, M., and Tomizawa, N. (1975 a): A unifying approach to fundamental problems in network theory by means of matroids (in Japanese). Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. 58 A, pp. 33–40.

    Google Scholar 

  • Iri, M., and Tomizawa, N. (1975 b): An algorithm for finding an optimal “independent assignment”. Journal of the Operations Research Society of Japan, Vol. 19, pp. 32–57.

    Google Scholar 

  • Iri, M., Tomizawa, N., and Fujishige, S. (1977): On the controllability and observability of a linear system with combinatorial constraints (in Japanese). Transactions of the Society of Instrument and Control Engineers, Japan, Vol. 13, pp. 225–242.

    Google Scholar 

  • Iri, M., Tsunekawa, J., and Murota, K. (1982): Graph-theoretic approach to large-scale systems - Structural solvability and block-triangularization (in Japanese). Transactions of Information Processing Society of Japan, Vol. 23, pp. 88–95.

    Google Scholar 

  • Iri, M., Tsunekawa, J., and Yajima, K. (1971): The graphical techniques used for a chemical process simulator “JUSE GIFS”. Information Processing 71 (Proceedings of the 1971 IFIP Congress), Vol. 2 (Applications), pp. 1150–1155.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Kel’mans, A. K., Lomonosov, N. V., and Polesskii, V. P. (1976): Minimum matroid coverings (in Russian). Problemy Peredachi Informatsii, Vol. 12, pp. 94–107.

    MathSciNet  MATH  Google Scholar 

  • Kishi, G., and Kajitani, Y. (1968): Maximally distinct trees in a linear graph (in Japanese). Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. 51 A, pp. 196–203.

    Google Scholar 

  • Knuth, D. E. (1973): Matroid partitioning. Report Stan-CS-73-342, Stanford University.

    Google Scholar 

  • Korte, B., and Hausmann, D. (1978): An analysis of the greedy heuristic for independence systems. Annals of Discrete Mathematics, Vol. 2, pp. 65–74.

    Article  MathSciNet  MATH  Google Scholar 

  • Kron, G. (1939): Tensor Analysis of Networks. John Wiley and Sons, New York.

    Google Scholar 

  • Kron, G. (1963): Diakoptics - The Piecewise Solution of Large-Scale Systems. Mac- donald, London.

    Google Scholar 

  • Kuh, E. S., Layton, D., and Tow, J. (1968): Network analysis via state variables. In Network and Switching Theory ( Biorci, G., ed.), Academic Press, New York.

    Google Scholar 

  • Kuh, E. S., and Rohrer, R. A. (1965): The state variable approach to network analysis. Proceedings of the IEEE, Vol. 53, pp. 672–686.

    Article  Google Scholar 

  • Kung, J. P. S. (1978): Bimatroids and invariants. Advances in Mathematics, Vol. 30, pp. 238–249.

    Article  MathSciNet  MATH  Google Scholar 

  • Laman, G. (1970): On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics, Vol. 4, pp. 331–340.

    Article  MathSciNet  MATH  Google Scholar 

  • Lawler, E. L. (1975): Matroid intersection algorithms. Mathematical Programming, Vol. 9, pp. 31–56.

    Article  MathSciNet  MATH  Google Scholar 

  • Lawler, E. L. (1976): Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York.

    Google Scholar 

  • Lehman, A. (1964): A solution to the Shannon switching game. Journal of the Society of the Industrial and Applied Mathematics, Vol. 12, pp. 687–725.

    Article  MathSciNet  MATH  Google Scholar 

  • Lovász, L. (1977): Flats in matroids and geometric graphs. In Combinatorial Surveys (Cameron, P., ed.), (Proceedings of the 6th British Combinatorial Conference), Academic Press, pp. 45–86.

    Google Scholar 

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

    MATH  Google Scholar 

  • Lovász, L. (1981): The matroid matching problem. In Algebraic Methods in Graph Theory (Lovász, L., and Sós, V. T., eds.), Colloquia Mathematica Societatis Jänos Bolyai 25, North-Holland, pp. 495–517.

    Google Scholar 

  • Lovász, L., and Yemini, Y. (1982): On generic rigidity in the plane. SIAM Journal on Algebraic and Discrete Methods, Vol. 3, pp. 91–98.

    Article  MATH  Google Scholar 

  • Manabe, R., and Kotani, S. (1973): On the minimal spanning arborescence of a direct graph (in Japanese). Keiei-Kagaku (Official Journal of the Operations Research Society of Japan), Vol. 17, pp. 269–278.

    MathSciNet  Google Scholar 

  • Maurer, S. B. (1975): A maximum-rank minimum-term-rank theorem for matroids. Linear Algebra and Its Applications, Vol. 10, pp. 129–237.

    Article  MathSciNet  MATH  Google Scholar 

  • Meetz, K., and Engl, W. L. (1980): Electromagnetische Felder. Springer-Verlag, Berlin.

    Google Scholar 

  • Megiddo, N. (1974): Optimal flows in networks with multiple sources and sinks. Mathematical Programming, Vol. 7, pp. 97–107.

    Article  MathSciNet  MATH  Google Scholar 

  • McDiarmid, C. J. H. (1975): Rado’s theorem for polymatroids. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 78, pp. 263–281.

    Article  MathSciNet  MATH  Google Scholar 

  • Milic, M. (1974): General passive networks: solvability, degeneracies and order of complexity. IEEE Transactions on Circuits and Systems, Vol. CAS-21, pp. 173–183.

    Google Scholar 

  • Minty, G. J. (1966): On the axiomatic foundations of the theories of directed linear graphs, electrical networks and network programming. Journal of Mathematics and Mechanics, Vol. 15, pp. 485–520.

    MathSciNet  MATH  Google Scholar 

  • Mizoo, Y., Iri, M., and Kondo, K. (1958): On the torsion and linkage characteristics and the duality of electric, magnetic and dielectric networks. RAAG Memoirs, Vol. 2, A- VIII, pp. 84–117.

    Google Scholar 

  • Murota, K. (1982): Menger-decomposition of a graph and its application to the structural analysis of a large-scale system of equations. Kokyuroku 453, Research Institute of Mathematical Sciences, Kyoto University, pp. 127–173.

    Google Scholar 

  • Murota, K., and Iri, M. (1982): Matroidal approach to the structural solvability of a system of equations. XIth International Symposium on Mathematical Programming, Universität Bonn.

    Google Scholar 

  • Nakamura, M. (1982 a): Boolean sublattices connected with minimization problems on matroids. Mathematical Programming, Vol. 22, pp. 117–120.

    Google Scholar 

  • Nakamura, M. ( 1982 b): Mathematical Analysis of Discrete Systems and Its Applications (in Japanese). Doctor’s dissertation, Faculty of Engineering, University of Tokyo.

    Google Scholar 

  • Nakamura, M., and Iri, M. (1979): Fine structures of matroid intersections and their applications. Proceedings of the International Symposium on Circuits and Systems, Tokyo, pp. 996–999.

    Google Scholar 

  • Nakamura, M., and Iri, M. ( 1980 a): On the decomposition of a directed graph with respect to arborescences and related problems. Kokyuroku 396 (Symposium on Graphs and Combinatorics III), Research Institute of Mathematical Sciences, Kyoto University, pp. 104–118.

    Google Scholar 

  • Nakamura, M., and Iri, M. (1980 b): Polylinking systems and their principal partition (in Japanese). Proceedings of the 1980 Spring Conference of the Operations Research Society of Japan, B-6, pp. 56–57.

    Google Scholar 

  • Nakamura, M., and Iri, M. (1981): A structural theory for submodular functions, polymatroids and polymatroid intersections. Research Memorandum RMI 81 - 06, Department of Mathematical Engineering and Instrumentation Physics, University of Tokyo.

    Google Scholar 

  • Narayanan, H. (1974): Theory of Matroids and Network Analysis. Ph. D. Thesis, Department of Electrical Engineering, Indian Institute of Technology, 117, Bombay.

    Google Scholar 

  • Narayanan, H., and Vartak, M. N. (1981): An elementary approach to the principal partition of a matroid. Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. E 64, pp. 227–234.

    Google Scholar 

  • Numata, J., and Iri, M. (1973): Mixed-type topological formulas for general linear networks. IEEE Transactions on Circuit Theory, Vol. CT-20, pp. 488–494.

    Google Scholar 

  • Ohtsuki, T., Ishizaki, and Watanabe, H. (1968): Network analysis and topological degrees of freedom (in Japanese). Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. 51 A, pp. 238 - 245.

    Google Scholar 

  • Oono, Y. (1960): Formal readability of linear networks. Proceedings of the Symposium on Active Networks and Feedback Systems, Polytechnic Institute of Brooklyn, New York, pp. 475–586.

    Google Scholar 

  • Ozawa, T. (1972): Order of complexity of linear active networks and common tree in the 2-graph method, Electronics Letters, Vol. 8, pp. 542–543.

    Article  MathSciNet  Google Scholar 

  • Ozawa, T. (1974): Common trees and partition of two-graphs (in Japanese). Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. 57 A, pp. 383–390.

    Google Scholar 

  • Ozawa, T. (1976 a): Topological conditions for the solvability of linear active networks. Circuit Theory and Applications, Vol. 4, pp. 125–136.

    Google Scholar 

  • Ozawa, T. (1976 b): Structure of 2-graphs (in Japanese). Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. J 59 A, pp. 262–263.

    Google Scholar 

  • Ozawa, T., and Kajitani, Y. (1979): Diagnosability of linear active networks. Proceedings of the International Symposium on Circuits and Systems, Tokyo, pp. 866–869.

    Google Scholar 

  • Petersen, B. (1978): Investigating solvability and complexity of linear active networks. Proceedings of the 1978 European Conference on Circuit Theory and Design, Lausanne.

    Google Scholar 

  • Petersen, B. (1979 a): Investigating solvability and complexity of linear active networks by means of matroids. IEEE Transactions on Circuits and Systems. Vol. CAS-26, No. 5, pp. 330–342.

    Google Scholar 

  • Petersen, B. ( 1979 b): The qualitative appearance of linear active network transfer functions by means of matroids. Proceedings of the International Symposium on Circuits and Systems, Tokyo, pp. 992–995.

    Google Scholar 

  • Tomizawa, N. (1976): Strongly irreducible matroids and principal partitions of a matroid into strongly irreducible minors (in Japanese). Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. J 59 A, pp. 83–91.

    Google Scholar 

  • Tomizawa, N., and Fujishige, S. (1982): Historical survey of extensions of the concept of principal partition and their unifying generalization to hypermatroids. Systems Science Research Report, No. 5, Department of Systems Science, Graduate School of Science and Engineering, Tokyo Institute of Technology.

    Google Scholar 

  • Tomizawa, N., and Iri, M. (1974): An algorithm for determining the rank of a triple matrix product A XB with application to the problems of discerning the existence of the unique solution in a network (in Japanese). Transactions of the Institute of Electronics and Communication Engineers of Japan, Vol. 57 A, pp. 834–841.

    Google Scholar 

  • Tomizawa, N., and Iri, M. (1976): Matroids (in Japanese). Journal of the Institute of Electronics and Communication Engineers of Japan, Vol. 59, pp. 1350 - 1352.

    Google Scholar 

  • Tomizawa, N., and Iri, M. (1977): On matroids (in Japanese). Journal of the Society of Instrument and Control Engineers, Japan, Vol. 16, pp. 455–468.

    Google Scholar 

  • Weinberg, L. (1962): Network Analysis and Synthesis. McGraw-Hill.

    Google Scholar 

  • Weinberg, L. (1977): Matroids, generalized networks, and electric network synthesis. Journal of Combinatorial Theory, Series B, Vol. 23, pp. 106–126.

    MathSciNet  MATH  Google Scholar 

  • Welsh, D. J. A. (1968): Kruskal’s theorem for matroids. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 64, pp. 3 - 4.

    Article  MathSciNet  MATH  Google Scholar 

  • Welsh, D. J. A. (1970): On matroid theorems of Edmonds and Rado. Journal of the London Mathematical Society, Vol. 2, pp. 251–256.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Whitney, H. (1935): On the abstract properties of linear dependence. American Journal of Mathematics, Vol. 57, pp. 509–533.

    Article  MathSciNet  Google Scholar 

  • Wilson, R. J. (1973): An introduction to matroid theory. American Mathematical Monthly, Vol. 80, pp. 500–525.

    Article  MathSciNet  MATH  Google Scholar 

  • Yajima, K., Tsunekawa, J., and Kobayashi, S. (1981): On equation-based dynamic simulation. Proceedings of the 2 nd World Congress of Chemical Engineering, Vol. V, pp. 469–480.

    Google Scholar 

  • Yao, A. C.C. (1975): An O(|E|log log|V|) algorithm for finding minimum spanning trees. Information Processing Letters, Vol. 4, pp. 21 - 25.

    Article  MATH  Google Scholar 

  • Zimmermann, U. (1981a): Minimization of some nonlinear functions over polymatroidal flows. Report 81 - 5, Mathematisches Institut, Universität zu Köln.

    Google Scholar 

  • Zimmermann, U. (1981b): Minimization on submodular flows. Discrete Applied Mathematics, Vol. 4, pp. 303–323.S

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Engineering, Dept. of Mathematical Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113, Japan

    M. Iri

Authors
  1. M. Iri
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Ökonometrie und Operations Research, Universität Bonn, Nassestraße 2, D-5300, Bonn 1, Germany

    Achim Bachem  & Bernhard Korte  & 

  2. Lehrstuhl für Angewandte Mathematik II, Universität Augsburg, Memminger Straße 6, D-8900, Augsburg, Germany

    Martin Grötschel

Rights and permissions

Reprints and permissions

Copyright information

© 1983 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Iri, M. (1983). Applications of Matroid Theory. In: Bachem, A., Korte, B., Grötschel, M. (eds) Mathematical Programming The State of the Art. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-68874-4_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-68874-4_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-68876-8

  • Online ISBN: 978-3-642-68874-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation