Edge-connectivity of undirected and directed hypergraphs

76 Chapter 4. Connectivity augmentation of directed hypergraphs
4.3.1 Adding hyperarcs of minimum total size
Theorem 4.8. Let p : 2 V → Z+ be a positively crossing supermodular set function. The
minimum total size σ of a directed hypergraph that covers p is σ = σ1 + σ2, where
1
σ1 = max max p(X),
p(V − X) : F a partition of V ,
|F| − 1
Z∈F
Z∈F
σ2 = max max p(X),
p(V − X) : F a partition of V .
Z∈F
Z∈F
Proof. To see the necessity of the conditions, let D = (V, A) be a directed hypergraph of
minimum total size that covers p, and let mi(v) := ϱD(v) and mo(v) := δD(v) for every
v ∈ V . Then mi(X) ≥ p(X) and mo(V − X) ≥ p(X) for every X ⊆ V . Let F be
a partition of V . Then mi(V ) ≥
Z∈F p(X), and (|F| − 1)mi(V ) ≥
Z∈F p(V − X).
Similarly, mo(V ) ≥
Z∈F p(V − X), and (|F| − 1)mo(V ) ≥
Z∈F p(X); in addition, we
know that mo(V ) ≥ mi(V ). We obtained that mi(V ) ≥ σ1 and mo(V ) ≥ σ2, thus the total
size of D is at least σ.
Sufficiency is proved by finding degree specifications mi and mo for which we can apply
the complete splitting-off described in Theorem 4.7. So we have to find mi satisfying (4.5)
and mo satisfying (4.6) such that mi(V ) ≤ mo(V ).
Let p1 be the set function defined by p1(X) := p(X) if X ⊂ V and p1(V ) := σ1. If the
conditions of the theorem hold, p1 is positively crossing supermodular, and by Theorem
2.13 B(p1) is a non-empty base polyhedron. Let mi be an integral vector in B(p1); then
mi satisfies (4.5).
Let p2 be defined by p2(X) = p(V −X) if ∅ = X ⊂ V , and p2(V ) := σ2. If the conditions
of the theorem hold, then p2 is positively crossing supermodular, and B(p2) is a non-empty
base polyhedron by Theorem 2.13. Let mo be an integral vector in B(p2); then mo satisfies
(4.6) and mi(V ) = σ1 ≤ σ2 = mo(V ).
We can apply Theorem 4.7 on these mi and mo values, and a suitably chosen r.
4.3.2 Covering by uniform directed hypergraphs
We can use the degree-specified result in Theorem 4.5 to give a formula on the minimal
number of (r, 1)-hyperedges that can cover a given positively crossing supermodular set
function.
Theorem 4.9. Let p : 2 V → Z+ be a positively crossing supermodular set function, and r
a positive integer. There exists a directed (r, 1)-hypergraph with γ hyperarcs that covers p