Abstract
Letf be a submodular function on 2E for a nonempty finite setE and λ be a parameter that takes on values between −∞ and ∞. In this paper, we show that the set of the subpartitions π ofE on which ΣX ∈Π(f - λ)(X) attains the minimum has a structure similar to that of the intersecting family. Moreover, we apply this result to the minimum augmentation problem with respect to the edge-connectivity. We reveal that when a given graphG=(V, A) is notK-edge-connected, the set of the subpartitions π of the vertex set except {V} on which ΣX ∈Π(d -K)(X) attains the minimum has a structure like the cointersecting family, whered(X) denotes the number of edges betweenX andV−X.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Cai and Y. Sun, The minimum augmentation of any graph to aK-edge-connected graph. Networks,19 (1989), 151–172.
A. Frank, On disjoint trees and arborescences Algebraic Methods in Graph Theory (eds. L. Lóvasz and V. T. Sós), North-Holland, Amsterdam, 1978, pp. 159–169.
A. Frank, Augmenting graphs to meet edge-connectivity requirements. SIAM J. Discrete Math.,5 (1992), 25–53.
A. Frank, Submodular functions in graph theory. Discrete Math.,111 (1993), 231–243.
T. Naitoh, Studies in Combinatorial Optimization Related to Submodular and Bisubmodular Functions. Dissertation. Doctoral Program in Socio-Economic Planning, University of Tsukuba, December, 1993.
H. Narayanan, The principal lattice of partitions of a submodular function. Linear Algebra Appl.,144 (1991), 179–216.
Author information
Authors and Affiliations
About this article
Cite this article
Naitoh, T., Nakayama, A. Structures of subpartitions related to a submodular function minimization. Japan J. Indust. Appl. Math. 14, 25–32 (1997). https://doi.org/10.1007/BF03167307
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03167307