Edge-connectivity of undirected and directed hypergraphs

122 Chapter 7. Combined augmentation and orientation
Throughout the process, F1 and F2 are compositions. The cardinality of the set for
which they are both compositions decreases in operation (iii), while operations (iv) and
(v) decrease either hX(F1) or hX(F2). Thus by Lemma 2.4 the process terminates after
a finite number of steps. We denote by F ′ 1 and F ′ 2 the families obtained at the end. It is
easy to see that F ′ 1 and F ′ 2 are tree-compositions of some X ′ ⊆ X, F ′ 1 + F ′ 2 is cross-free,
and hX ′(F ′ 2) ≤ 0 by Claim 2.2. Moreover, QX(F1, F2) does not decrease in any of the
steps (in (i)–(iii) this follows from the supermodularity of p, in (iv) it is a consequence of
Claim 7.6, and in (v) it can be deduced from the definition of QX(F1, F2)). This proves
that maxX⊆V q ′ (X) = maxX⊆V q(X).
By Claim 7.8, σ ≥ maxX⊆V q ′ (X). Thus there exists a vector m : V → Z+ with
m(V ) = σ that satisfies (7.1)–(7.3), therefore by Theorem 7.1 there exists a hypergraph H
with γ hyperedges of total size σ such that H0 + H has an orientation covering p, and H
satisfies (7.15). This concludes the proof of Theorem 7.5.
If the requirement function p is monotone decreasing or symmetric, then Theorem 7.5
can be simplified.
Theorem 7.9. Let H0 = (V, E0) be a hypergraph, p : 2 V → Z+ a monotone decreasing
or symmetric non-negative crossing supermodular set function, σ ≥ 0 and 0 ≤ γ ≤ σ/2
integers. There exists a hypergraph H with γ hyperedges of total size σ such that H0 + H
has an orientation covering p if and only if the following hold:
γ ≥
p(Z) − eH0(F) for every partition F, (7.17)
σ ≥
Z∈F
Z∈F1+co(F2)
p(Z) − eH0 (F1 + co(F2)) (7.18)
whenever F1 and F2 are partitions of some X ⊆ V and F1 is a refinement of F2. In
addition, H can be chosen so that
σ
≤ |e| ≤
γ
σ
γ
for every e ∈ E. (7.19)
Proof. It suffices to show that if condition (7.14) is violated for some pair (F1, F2), then it is
also violated by a pair (F ′ 1, F ′ 2) that has the additional properties that F ′ 1 is a subpartition,
and Y2 ⊂ Y1 for every Y1 ∈ F ′ 1, Y2 ∈ F ′ 2. Such families can be obtained from F1 and F2 by
repeating the following operations as long as any of them is possible:
(i) If F1 is a composition of Z ⊆ V , it contains a subfamily {W1, . . . , Ws} (s ≥ 2)
of pairwise co-disjoint sets such that W := ∩Wi ⊆ Z, and F2 contains a partition
{Z1, . . . , Zt} of W , then remove W1, . . . , Ws from F1 and Z1, . . . , Zt from F2.