Abstract
Letf be a submodular function on 2 E 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.
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