Edge-connectivity of undirected and directed hypergraphs

106 Chapter 6. Hypergraph orientation
of hyperarcs are called semi-parallel if every hyperarc in A ′ is an orientation of the same
hyperedge. In Theorem 6.13, we would obtain an equivalent problem if we replaced every
hyperedge of the mixed graph by a set of semi-parallel hyperarcs, consisting of all pos-
sible orientations of that hyperedge, and imposed the additional constraint that at most
one of these semi-parallel hyperarcs can be in the chosen sub-hypergraph. This concept
of orientation constraints can be further generalized: we allow arbitrary disjoint sets of
semi-parallel hyperarcs, and arbitrary lower and upper bounds on the number of hyperarcs
selectable from such a set.
Theorem 6.14. Let D = (V, A) be a directed hypergraph, with f : A → Z+ and g : A →
Z+ lower and upper integral capacities on the hyperarcs. Let A1, . . . , At ⊆ A be disjoint sets
of semi-parallel hyperarcs, with corresponding lower and upper bounds li, ui (i = 1, . . . , t).
Let furthermore p : 2 V → Z+ be a positively intersecting supermodular set function, and
c : A → Z a cost function. Then the system
(S) min
c(a)x(a) (6.7)
a∈A
ϱx(Z) ≥ p(Z) for every Z ⊆ V (6.8)
f(a) ≤ x(a) ≤ g(a) for every a ∈ A (6.9)
li ≤
x(a) ≤ ui (i = 1, . . . , t) (6.10)
a∈Ai
is TDI. Moreover, the values of an optimal dual solution corresponding to the inequalities
(6.8) may be assumed to be positive only on a laminar family of sets.
Proof. Let c : A → Z be an integral cost function. Let y denote the dual variables
associated with the inequalities in (6.8), and let z denote the dual variables associated
with the other inequalities. For a hyperarc a ∈ A, the dual constraints are of the form
⎛
⎝
⎞
y(X) ⎠ + zba ≤ c(a) (6.11)
X: a∈∆ −
D (X)
for an appropriate vector ba. For an appropriate vector b, the dual objective function is
max y(X)p(X) + zb . (6.12)
X⊆V
Let (y ∗ , z ∗ ) ≥ 0 be an optimal dual solution such that
Z⊆V y∗ (Z) is minimal. The
main observation is that we can assume that y ∗ is positive only on a laminar family F. If
p(X) = 0, then y ∗ (X) = 0, otherwise we could decrease y ∗ (X) to 0. Suppose that y ∗ is