Edge-connectivity of undirected and directed hypergraphs

Section 6.4. Directed network design problems with orientation constraints 105
v1 a1
v2 v3
v1 v2 v3 v4 v5 v6
v4
e1
a2
v5
e2
v6
Figure 6.2: Transformation into a submodular flow problem in Theorem 6.13
correspond to the hyperedges and hyperarcs in E ∪ A; we denote a node corresponding to
a hyperedge or hyperarc e by ue. The edge set B contains an edge from the head of e to ue
(with edge cost equal to the cost of e) if e is a hyperarc; if e is a hyperedge, then B contains
edges from every node of e to ue (each with cost equal to the cost of the corresponding
orientation of e). A set function q is defined on the ground set U ∪ V as follows:
⎧
⎪⎨ p(X ∩ V ) if ue ∈ X implies that the nodes of e are in X,
q(X) := −1
⎪⎩
−∞
if X = {ue} for some e ∈ E,
otherwise.
Figure 6.2 shows this construction. Let f ≡ 0 be the lower capacity of the edges and
g ≡ 1 the upper capacity. It is easy to see that if p is intersecting supermodular, then q is
crossing supermodular. So Theorem 2.28 implies that the problem of finding a minimum
ua1
cost directed subgraph D ′ = (U, V ; B ′ ) of D that satisfies
δD ′(X) − ϱD ′(X) ≥ q(X) for every X ⊆ U ∪ V (6.6)
can be solved in polynomial time (note that this is a one-way submodular flow system).
Since q({ue}) = −1 if e ∈ E, ue is the head of at most one edge of D ′ . Thus the subgraph
D ′ corresponds to an oriented sub-hypergraph M ′ of M. It is easy to check that M ′ covers
the requirement function p if and only if D ′ satisfies (6.6).
6.4.2 TDI property
The above construction shows that when the requirement function is intersecting supermod-
ular, the directed network design problem with orientation constraints can be transformed
into a submodular flow problem which is TDI. We now prove that the linear system that
we naturally associate to the original problem is also TDI, even for positively intersecting
supermodular requirement functions. To formulate the appropriate linear program, the
hypergraph analogue of orientation constraints must be defined. The elements of a set A ′
ua2
ue1
ue2
V
U