Edge-connectivity of undirected and directed hypergraphs

118 Chapter 7. Combined augmentation and orientation
Proof. By Theorem 2.11, it suffices to show that
pm(X) ≤ |E0| + γ, (7.9)
X∈F
pm(V − X) ≤ (|F| − 1) (|E0| + γ) (7.10)
X∈F
both hold for every partition F. Note that γ − m(V − X) can be positive for at most
one member of a partition, since γ ≤ m(V )/2. Thus (7.9) follows from (1.12), and either
(7.2) or (7.1), depending on whether F has such a member or not. The inequality (7.10)
follows from (1.12) and (7.3), if F ′ is chosen to consist of the members X of F for which
γ − m(X) ≤ 0.
By Theorem 2.11, B(pm) is a base polyhedron with integral vertices, and any such
vertex x is the vector of in-degrees (restricted to V ) of an orientation H ′ 0 of H ′ 0 satisfying
(7.5)–(7.7).
Let mi(v) be the multiplicity of the edge zv in H ′ 0, mo(v) be the multiplicity of the edge
vz in H ′ 0, and let H0 denote the directed hypergraph obtained from H ′ 0 by deleting the
node z. Then mi(X) ≥ p(X)−ϱ H0 (X) and mo(V −X) ≥ p(X)−ϱ H0 (X) for every X ⊆ V .
By (1.4) and the crossing supermodularity of p, the set function q(X) := p(X) − ϱH0 (X)
is crossing supermodular. Theorem 4.7 asserts the existence of a directed hypergraph
D = (V, A) that covers q, and satisfies the degree specifications mi and mo. This means
that H0 + D covers p, and the undirected hypergraph H that underlies D satisfies the
degree specification m. Theorem 4.7 also ensures that (7.4) is satisfied. Since H0 + D is
an orientation of H0 + H, this completes the proof of Theorem 7.1.
If the requirement function p is monotone decreasing or symmetric, then the conditions
of Theorem 7.1 can be simplified.
Theorem 7.4. Let H0 = (V, E0) be a hypergraph, p : 2 V → Z+ a monotone decreasing or
symmetric non-negative crossing supermodular set function, m : V → Z+ a degree speci-
fication and 0 ≤ γ ≤ m(V )/2 an integer. There exists a hypergraph H with γ hyperedges
satisfying the degree-specification m such that H0 + H has an orientation covering p if and
only if the following hold for every partition F of V :
γ ≥
p(Z) − eH0(F), (7.11)
Z∈F
min m(V − X) ≥ p(Z) − eH0(F). (7.12)
X∈F
Z∈F