Edge-connectivity of undirected and directed hypergraphs

40 Chapter 2. Submodular functions
According to Claim 2.22 we can assume that all tight sets are in G. If X and Y are
tight, then by Claim 2.20, 0 = p ∗ (Y ) + p ∗ (V − Y ) = p ∗ (X ∩ Y ) + p ∗ (Y − X) + p ∗ (V − Y ) ≤
p ∗ (X ∩ Y ) + p ∗ (V − (X ∩ Y ), so X ∩ Y is tight, and similarly Y − X is tight. This means
that the minimal tight sets form a partition W1, . . . , Wk. The conditions of the theorem
are not changed if we add a modular set function m to p for which m(V ) = 0. Thus we
can assume that p ∗ (W1) = · · · = p ∗ (Wk) = 0, so p ∗ (X) = 0 for every tight set X.
We say that sets X, Y are crossing restricted to Wl if X ∩ Wl and Y ∩ Wl are crossing
on the ground set Wl. For a given 1 ≤ l ≤ k we define a matrix Al. The rows of
Al correspond to the pairs X, Y ∈ G that are crossing restricted to Wl, and for which
p ∗ (X, Y ) = p ∗ (X) + p ∗ (Y ). The columns of Al correspond to the sets in G. In the
following we define the row of Al corresponding to a pair (X, Y ) of sets. There is a
composition FX∩Y of X ∩ Y and a composition FX∪Y of X ∪ Y , both consisting of sets
in G, for which p∗ (X, Y ) = p∗ (X ∩ Y ) + p∗ (X ∪ Y ) =
Z∈FX∩Y p∗ (Z) +
Z∈FX∪Y p∗ (Z).
Observe that FX∩Y + FX∪Y contains at least 2 non-tight sets (because Wl is a minimal
tight set), and if it contains exactly 2, then restricted to Wl these are (X ∩ Y ) ∩ Wl and
(X ∪ Y ) ∩ Wl. Using FX∩Y and FX∪Y , we define the row r of Al corresponding to X, Y :
for a set Z ∈ G, let r(Z) := −χFX∩Y (Z) − χFX∪Y (Z) if Z = X and Z = Y , and let
r(X) = r(Y ) := 1 − χFX∩Y − χFX∪Y (Z).
Claim 2.23. The system {yAl ≤ 0, y > 0} has no solution.
Proof. Suppose indirectly that there exists such a y; we can assume that y is integral.
The vector −yAl can be considered as a non-negative function whose domain of definition
is G. Let F be the family defined by χF(Z) := −yAl(Z) (Z ∈ A), and let F ′ be the
family obtained by leaving out the tight sets from F. By the definition of the rows of Al,
Z∈F p∗ (Z) = 0, and F is regular. This is possible only if F ′ = ∅. But in a row of Al
corresponding to a pair (X, Y ), exactly 2 columns corresponding to non-tight sets have
value +1 (those corresponding to X and Y ), and at least 2 columns corresponding to non-
tight sets have negative value. This is possible only if each row of Al for which y is positive
contains exactly 2 columns corresponding to non-tight sets that have negative value (which
is -1). We have seen that for a row corresponding to (X, Y ) the intersection of these non-
tight sets with Wl must be (X ∩ Y ) ∩ Wl and (X ∪ Y ) ∩ Wl. But |X ∩ Wl| 2 + |Y ∩ Wl| 2 <
|(X ∩ Y ) ∩ Wl| 2 + |(X ∪ Y ) ∩ Wl| 2 , so
Z∈F ′ |Z ∩ Wl| 2 > 0, contradicting F ′ = ∅.
According to the Lemma 1.16 the dual system {Alz ≫ 0, z ≥ 0} has a solution. Let zl
be an arbitrary solution of the above system (for l = 1, . . . , k). This means that zl is a
non-negative weight function on the sets of G, with the following property: