Abstract
This paper discusses the principal structure of submodular systems due to S. Fujishige. It is shown that the principal structure is the coarsest decomposition that is finer than any decomposition induced by the principal partition with respect to a minimal nonnegative superbase. The concept of “Hitchcock-type independent flow” is introduced so that previously known results on the principal structures for bipartite matchings, layered mixed matrices and independent matchings can be understood as applications of the present result.
Similar content being viewed by others
References
M. Aigner:Combinatorial Theory, Springer-Verlag, Berlin, 1979.
A. L. Dulmage andN. S. Mendelsohn: A structure theory of bipartite graphs of finite exterior dimension,Trans. Roy. Soc. Canada,53 (1959), 1–13.
A. Frank: An algorithm for submodular functions on graphs,Ann. Discrete Math.,16 (1982), 97–120.
S. Fujishige: Algorithms for solving the independent-flow problems.J. Oper. Res. Soc. Japan,21 (1978), 189–204.
S. Fujishige: Principal structures of submodular systems,Discrete Appl. Math.,2 (1980), 77–79.
S. Fujishige: Submodular systems and related topics,Math. Programming Stud.,22 (1984), 113–131.
S. Fujishige:Submodular Functions and Optimization, North-Holland, Amsterdam, 1991.
M. Iri: Applications of matroid theory,Mathematical Programming — The State of the Art (A. Bachem, M. Grötschel and B. Korte, eds.), Springer-Verlag, Berlin, 1983, 158–201.
M. Iri: Structural theory for the combinatorial systems characterized by submodular functions,Progress in Combinatorial Optimization (W. R. Pulleyblank, ed.), Academic Press, New York, 1984, 197–219.
S. Iwata andK. Murota: Horizontal principal structure of layered mixed matrices — Decomposition of discrete systems by design-variable selections,SIAM J. Discrete Math., to appear.
S. Iwata andK. Murota: A theorem on the principal structure for independent matchings,Discrete Appl. Math., to appear.
S. T. McCormick: A combinatorial approach to some sparse matrix problems,Tech. Rept. SOL 83-5, Department of Operations Research, Stanford University, 1983.
K. Murota:Systems Analysis by Graphs and Matroids — Structural Solvability and Controllability, Springer-Verlag, Berlin, 1987.
K. Murota: Principal structure of layered mixed matrices,Discrete Appl. Math.,27 (1990), 221–234.
K. Murota: Mixed matrices — irreducibility and decomposition,Combinatorial and Graph-Theoretical Problems in Linear Algebra (R. A. Brualdi, S. Friedland and V. Klee, eds.), The IMA Volumes in Mathematics and Its Applications, Springer-Verlag, 1993, 39–71.
K. Murota, M. Iri andM. Nakamura: Combinatorial canonical form of layered mixed matrices and its application to block-triangularization of systems of equations,SIAM J. Algebraic Discrete Methods,8 (1987), 123–149.
A. Recski:Matroid Theory and Its Applications in Electric Network Theory and in Statics, Springer-Verlag, Berlin, 1989.
N. Tomizawa andS. Fujishige: Theory of hyperspace (XIV) — Principal decompositions and principal structures of metric lattices with respect to supermodular functions (in Japanese),CAS 82-2, Institute of Electronics and Communications Engineers of Japan, 1982.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Iwata, S. Principal structure of submodular systems and hitchcock-type independent flows. Combinatorica 15, 515–532 (1995). https://doi.org/10.1007/BF01192524
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01192524