Edge-connectivity of undirected and directed hypergraphs

Section 7.2. Augmentation to meet orientability requirements 119
In addition, H can be chosen so that
m(V )
m(V )
≤ |e| ≤
γ
γ
for every e ∈ E. (7.13)
Proof. By definition, eH0(F) ≤ eH0(co(F)) for every partition F of V , and the monotonicity
or symmetry of p implies that
Z∈co(F) p(Z) ≤ Z∈F p(Z) also holds. It is easy to see
from this that (7.3) is implied by (7.1) if |F ′ | = 0 or |F ′ | ≥ 2, and it is implied by (7.2) if
|F ′ | = 1.
7.2.2 Minimum cardinality augmentation
As in the case of augmentation problems considered in Chapters 3 and 4, the charac-
terization of the degree specifications that allow a good augmentation helps to deduce a
characterization of the minimum number of hyperedges needed. In the present case we
obtain the following theorem:
Theorem 7.5. Let H0 = (V, E0) be a hypergraph, p : 2 V → Z+ a non-negative crossing
supermodular set function, σ ≥ 0 and 0 ≤ γ ≤ σ/2 integers. There exists a hypergraph
H = (V, E) with γ hyperedges of total size σ such that H0 + H has an orientation covering
p if and only if
σ + γhX(F1) + (σ − γ)hX(F2) ≥
Z∈F1+co(F2)
p(Z) − eH0 (F1 + co(F2)) (7.14)
whenever F1 and F2 are tree-compositions of some X ⊆ V , F1 + F2 is cross-free, and
hX(F2) ≤ 0 (i.e. either F2 is a partition of X, or X = V and F2 = ∅).
In addition, H can be chosen so that
σ
≤ |e| ≤
γ
σ
γ
for every e ∈ E. (7.15)
Proof. The right hand side of (7.14) is the deficiency of H0 on the family F1 + co(F2). The
number of sets of F1 that a new hyperarc enters is at most hX(F1), plus 1 if its head is in
X. The number of sets of co(F2) that a new hyperarc a enters is at most (|a| − 1)hX(F2)
plus the number of tail nodes it has in X. This shows the necessity of (7.14). To prove
sufficiency, we define the following functions for every X ⊆ V and compositions F1, F2 of
X:
QX(F1, F2) :=
Z∈F1+co(F2)
p(Z) − eH0 (F1 + co(F2)) − γhX(F1) − (σ − γ)hX(F2)),
q(X) := max{QX(F1, F2) : F1 and F2 are tree-compositions of X,
F1 + F2 is cross-free, hX(F2) ≤ 0}.