Edge-connectivity of undirected and directed hypergraphs

Section 6.4. Directed network design problems with orientation constraints 107
v1
a1
v2
v4
A1
a2
v3
v5
v6
A2
ϕ(v5)
ϕ(v1)
ϕ({v2, v4})
w1
ϕ(v3)
ϕ(v6)
Figure 6.3: Construction of the network matrix in Theorem 6.14
positive on two intersecting sets X and Y where p(X), p(Y ) > 0; let α = min{p(X), p(Y )}.
Decrease y ∗ (X) and y ∗ (Y ) by α, and increase y ∗ (X ∩ Y ) and y ∗ (X ∪ Y ) by α. Since
ϱa(X) + ϱa(Y ) ≥ ϱa(X ∩ Y ) + ϱa(X ∪ Y ) for each hyperarc a, the inequality (6.11) is
preserved. The positively intersecting supermodularity of p implies that the dual objective
function (6.12) does not decrease. By Claim 2.5, after a finite number of uncrossing steps,
we obtain an optimal dual solution where y ∗ is positive on a laminar family F.
Modify the system (S) by replacing (6.8) with
ϱx(Z) ≥ p(Z) for every Z ∈ F; (6.13)
let us denote this system by (S ′ ). Then (y ∗ , z ∗ ) remains a feasible dual solution, and it is
obviously optimal. Thus if the modified system has an integral optimal dual solution, it
is optimal for the dual of (S) as well. The rest of the proof consists of showing that the
system (S ′ ) can be described by a network matrix, hence it has an integral dual optimal
solution by Proposition 2.25.
The construction of the corresponding network is shown on Figure 6.3. The rows of
the network matrix will correspond to the edges of a directed tree T ′ = (W ′ , A ′ 1), and the
corresponding lower and upper bounds will be denoted by l ′ and u ′ . The laminar family F
has a tree-representation (T, ϕ) where T = (W, A1) is an arborescence; T ′ will include T as
a subtree. For an edge e ∈ A1 let l ′ (e) = −∞ and u ′ (e) = −p(ϕ −1 (We)), where We is the
component of T − e entered by e. The node set W ′ is obtained by adding new nodes wi
(i = 1, . . . , t) to W (that is, one new node wi for each orientation constraint set Ai which
consists of semi-parallel hyperedges). For a set Z ⊆ V let wZ ∈ W denote the root node of
the minimal subtree of T containing all nodes of ϕ(Z). To finish the construction of T ′ , add
an edge ei = wZi wi to A ′ 1 for i = 1, . . . , t, where Zi is the node set of the hyperedge whose
orientations are in Ai. Define the corresponding lower and upper bounds as l ′ (ei) = li,
u ′ (ei) = ui.
w2