Abstract
This paper presents a lattice-theoretic decomposition method for submodular functions. This decomposition is derived from a lattice, called a ‘skeleton’, which has the property that the specified submodular function is reduced to be modular on it.
Several types of skeletons arise from the various kinds of minimization problems of submodular functions. The minimization problem associated with the well-known min-max eqaulity of the polymatroid intersection problem affords a skeleton, which is shown to provide a direct-sum decomposition of the maximum common independent vectors, i.e. the solutions of the intersection problem. Furthermore, a parametrized polymatroid intersection problem is considered and the construction of its solution, called a ‘universal pair’, is described. This paper consists of the main results of the author's unpublished works [27], [35].
Similar content being viewed by others
References
Aigner, M.: Combinatorial Theory. Berlin: Springer-Verlag 1979
Birkhoff, G.: Lattice Theory (third edition). Amer. Math. Soc. Colloquim Pub., Providence, 1967
Bruno, J., Weinberg, L.: A constructive graph-theoretic solution of the Shannon switching game. IEEE Trans. on Circuit Theory.17, 74–81 (1970)
Bruno, J., Weinberg, L.: The principal minors of a matroid. Linear Algebra Appl.4, 17–54 (1971)
Dulmage, A.L., Mendelsohn, N.S.: Coverings of bipartite graphs. Canad. J. Math.,10, 517–534 (1958)
Dulmage, A.L., Mendelsohn, N.S.: A structure theory of bipartite graphs of finite exterior dimension. Trans. Royal Society of Canada, Third Series Section III,53, 1–13 (1959)
Dulmage, A.L., Mendelsohn, N.S.: On the inversion of sparse matrices. Math. Comput.16, 494–496 (1962)
Dulmage, A.L., Mendelsohn, N.S.: Two algorithms for bipartite graphs. J. SIAM.11, 183–194 (1963)
Edmonds, J.: Minimum partition of a matroid into independent subsets. J. Res. Natl. Bur. Stand.69B, 73–77 (1965)
Edmonds, J.: Submodular functions, matroids and certain polyhedra. In: Combinatorial Structure and Their Applications. pp. 69–87. New York: Gordon and Breach 1970
Edmonds, J., Fulkerson, D.R.: Transversals and matroid partition. J. Res. Natl. Bur. Stand.69B, 147–157 (1965)
Edmonds, J., Giles, R.: A min-max relation for submodular functions on graphs. Ann. Discrete Math.1, 185–204 (1977)
Fujishige, S.: Algorithms for solving the independent-flow problems. J. Operations Research Soc. Japan21, 189–203 (1978)
Frank, A.: Generalized polymatroids. In: Finite and Infinite Sets I, edited by A. Haynal, L. Lovasz, V.T. Sos. pp. 285–294. Amsterdam: North-Holland 1984
Iri, M.: Combinatorial canonical form of a matrix with applications to the principal partition of a graph (in Japanese). Trans. IECEJ54A, 30–37 (1971)
Iri, M.: A reivew of recent work in Japan on principal partitions of matroids and their applications. Annals of the New York Academy of Sciences (Proceedings of the Second International Conference on Combinatorial Mathematics).319, 306–319 (1979)
Iri, M.: Structural theory for the combinatorial systems characterized by submodular functions. In: Progress in Combinatorial Optimization, edited by W.R. Pulleyblank. pp. 179–219. New York: Academic Press 1984
Iri, M., Tomizawa, N.: An algorithm for finding an optimal ‘independent assignment’. J. Oper. Res. Soc. Jpn19, 32–57 (1976)
Kel'mans, A.K., Lomonosov, N.V., Polesskii, V.P.: Minimum matroid coverings. Problemy Peredachi In.12, 94–107 (1976)
Kishi, G., Kajitani, Y.: Maximally distinct trees. In: Proc. Fifth Annual Allerton Conference on Circuit and System Theory. pp. 635–643. 1967
Kishi, G., Kajitani, Y.: Maximally distinct trees in a linear graph (in Japanese). Trans. IECEJ51A, 196–203 (1968)
Kung, J.P.S.: Bimatroids and invariants. Advances in Math.30, 238–249 (1978)
Lawler, E.L.: Matroid intersection algorithms. Math. Program.9, 31–56 (1975)
Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. New York: Holt, Rinehart and Winston 1976
Lehman, A.: A solution to the Shannon switching game. J. Soc. Industrial and Applied Math.,12, 687–725 (1964)
Meggido, N.: Optimal flows in network with multiple sources and sinks. Math. Program.7, 97–107 (1974)
Nakamura, M.: Mathematical analysis of discrete systems and its applications (in Japanese). Doctoral Thesis. Dept. of Mathematical Engineering, University of Tokyo 1982
Nakamura, M.: Boolean sublattices connected with minimization problems on matroids. Math. Program.22, 117–120 (1982)
Nakamura, M.: Analysis of discrete systems and its applications (in Japanese). Trans. IECEJ66A, 368–373 (1983)
Nakamura, M.: On the representation of the rigid sub-systems of a plane link system. J. Operations Research Soc. Jpn29, 305–318 (1986)
Nakamura, M.: A note on the decomposition of poly-linking systems and the minors of generalized polymatroids (in preparation)
Nakamura, M., Iri, M.: Fine structures of matroid intersections and their applications. In: Proc. Int. Symp. Circuits and Systems, Tokyo. pp. 996–999. 1979
Nakamura, M., Iri, M.: On the decomposition of a directed graph with respect to arborescences and related problems. In: R.I.M.S. Symposium on Graphs and Combinatorics III,397. pp. 104–118. September 1980
Nakamura, M., Iri, M.: Poly-linking systems and the principal partition. In: Proc. Spring Symp. Operations Research Soc. Jpn. pp. 56–57. 1980
Nakamura, M., Iri, M.: A structural theory for submodular functions, polymatroids and polymatroid intersections. In: Research Memorandum RMI 81-06. Dept. of Math. Engineering, Univ. of Tokyo 1981
Narayanan, H.: Theory of Matroids and Network Analysis. Doctoral thesis. Dept. Electrical Engineering, Indian Institute of Technology, Bombay. February 1974
Nash-Williams, C. St. J.A.: Edge-disjoint spanning trees on finite graphs. J. London Math. Soc.36, 445–450 (1961)
Nash-Williams, C. St. J.A.: Decomposition of finite graphs into forests. J. London Math. Soc.39, 12 (1964)
Ohtsuki, T., Tsuchiya, T., Ishizaki, Y., Watanabe, H., Kajitani, Y. Kishi, G.: Topological degrees of freedom of electrical networks. In: Proc. Fifth Annual Allerton Conference on Circuit and System Theory. pp. 644–653. 1967
Ohtsuki, T., Ishizaki Y., Watanabe, H.: Network analysis and topological degrees of freedom (in Japanese). Trans. IECEJ51A, 238–245 (1968)
Picard, J.C., Queyranne, M.: On the structure of all minimum cuts in a network and applications. Math. Program.13, 8–16 (1980)
Polesskii, V.P.: Isthmus structure in a summary matroid. Problemy Peredachi Inf.12, 95–104 (1976)
Schrijver, A.: Matroids and Linking Systems. Amsterdam: Mathematical Centre Tracts 88 1978
Sugihara, K.: Studies on mathematical structures of line drawings of polyhedra and their applications to scence analysis (in Japanese). In: Researches of the Electrotechnical Laboratory, No. 800. (1979)
Tomizawa, N.: Irreducible matroids and principal partitions of a matroid into irreducible minors (in Japanese). In: Papers of the Technical Group on Circuit and System Theory of the IECEJ. pp. 74–8. (1974)
Tomizawa, N.: Strongly irreducible matroids and principal partitions of a matroid into strongly irreducible minors (in Japanese). Trans. IECEJ59A, 83–91 (1976)
Tomizawa, N.: Theory of hyperspaces I–VII (in Japanese). In: Papers of the Technical Group on Circuit ans System Theory of the IECEJ, Vol. CAS 80-72, 74, 75, 85, 95, 96, 106. (1980)
Tutte, W.T.: On the problem of decomposing a graph into n connected factors. J. London Math. Soc.36, 221–230 (1961)
Welsh, D.J.A.: Matroid Theory. London: Academic Press 1976
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nakamura, M. Structural theorems for submodular functions, polymatroids and polymatroid intersections. Graphs and Combinatorics 4, 257–284 (1988). https://doi.org/10.1007/BF01864166
Issue Date:
DOI: https://doi.org/10.1007/BF01864166