Edge-connectivity of undirected and directed hypergraphs

34 Chapter 2. Submodular functions
2.2.4 Intersecting and crossing supermodularity
For set functions that appear in connectivity problems, the supermodular inequality (2.2)
often does not hold for every pair of sets. For example, given a hypergraph H0 and a
positive integer k, let p(X) := k − dH0(X) if ∅ = X ⊂ V , and p(∅) := p(V ) := 0. For a
hypergraph H, H0 + H is k-edge-connected if and only if H covers p. The set function p
is not necessarily supermodular on non-crossing sets.
Another example is for a given directed hypergraph D0 with a fixed root node s: let
p(X) := k −ϱD0(X) is ∅ = X ⊆ V −s, and p(X) := 0 otherwise. For a directed hypergraph
D, D0 + D is k-rooted-connected from s if and only if D covers p. Here p is supermodular
on intersecting sets.
This shows that in many cases the requirement of full supermodularity must be relaxed.
A set function p : 2 V → Z ∪ {−∞} is crossing supermodular if (2.2) holds whenever X and
Y are crossing. The set function p is intersecting supermodular if (2.2) holds whenever X
and Y are intersecting.
For crossing supermodular functions, the following theorem of Fujishige characterizes
the non emptiness of B(p).
Theorem 2.11 (Fujishige [38]). Let p : 2 V → Z ∪ {−∞} be a crossing supermodular
function. Then B(p) is nonempty if and only if
t
p(Xi) ≤ p(V ),
i=1
t
p(V − Xi) ≤ (t − 1)p(V )
i=1
both hold for every partition {X1, X2, . . . , Xt} of V . Furthermore, if B(p) is non-empty,
then it is a base polyhedron.
If p is intersecting supermodular, then the non-emptiness condition reduces to the fol-
lowing:
Proposition 2.12. Let p : 2 V → Z ∪ {−∞} be an intersecting supermodular function.
Then B(p) is nonempty if and only if
t
p(Xi) ≤ p(V )
i=1
holds for every partition {X1, X2, . . . , Xt} of V . Furthermore, if B(p) is non-empty, then
it is a base polyhedron.