Edge-connectivity of undirected and directed hypergraphs

36 Chapter 2. Submodular functions
The full truncation of a set function p : 2 V → Z ∪ {−∞} is
p ↑
(X) := max p(Z) − hX(F)p(V ) : F is a tree-composition of X . (2.11)
Z∈F
Here B(p) = B(p ↑ ), and the following is true:
Proposition 2.15 (Frank [25]). If p is crossing supermodular, and B(p) is non-empty,
then p ↑ is fully supermodular.
The algorithms to calculate the upper truncation and the full truncation will be briefly
discussed in Section 2.5. Another type of truncation appears when we define a matroid
using an intersecting submodular set function. Edmonds [15] proved the following theorem:
Theorem 2.16 (Edmonds [15]). Let b : 2 S → Z+ be a non-negative, integer-valued,
intersecting submodular set-function. Then
Ib := {X ⊆ S : b(Y ) ≥ |Y ∩ X| for every Y ⊆ S} (2.12)
forms the family of independent sets of a matroid Mb = (S, Ib) whose rank-function is given
by
rMb (X) = min b(Z) + |X − (∪Z∈FZ)| : F is a subpartition of S . (2.13)
Z∈F
Furthermore, if b is monotone increasing, then
and
Ib = {X ⊆ S : b(Y ) ≥ |Y | for every Y ⊆ X} (2.14)
rMb (X) = min b(Z) + |X − (∪Z∈FZ)| : F is a subpartition of X . (2.15)
Z∈F
2.3 Relaxations of supermodularity
In this section we discuss how supermodularity can be further relaxed while some other
important properties, like the TDI property of the associated linear system, are retained.
Theorem 2.17 is a comprehensive result on possible relaxations that seems to have not been
observed before. The other new observation of the section, Theorem 2.24, describes a way
to relax the conditions of Theorem 2.9.