Edge-connectivity of undirected and directed hypergraphs

30 Chapter 2. Submodular functions
Proof. This well-known result can be seen as a special case of the following claim (note
that the claim does not hold for non-negative real numbers!):
Claim 2.5. Let x1, . . . , xn be non-negative rational numbers. Suppose that we apply re-
peatedly the following operation: for some indices i < j < k < l where xj and xk are
positive, decrease xj and xk by min{xj, xk}, and increase xi and xl by min{xj, xk}. Then
this operation can be repeated only a finite number of times.
Proof. By multiplying all xi values by a suitable integer, we can assume that every xi is
integer. Now suppose that there is an infinite sequence of operations, and let m be the
smallest index for which xm decreases infinitely many times. Then one of x1, . . . , xm−1
increases infinitely many times by at least 1, but decreases finitely many times, which is
impossible since n
i=1 xi remains constant and xi ≥ 0 for every i.
Let X1, . . . , Xt be an ordering of the subsets of V compatible with the standard partial
order; let xi := y(Xi). Then it follows from the claim that after finitely many uncrossing
steps uncrossing is impossible, therefore y is positive on a cross-free family.
To illustrate the usefulness of the uncrossing technique, let p : 2 V → Z ∪ {−∞} be a set
function for which p(X) + p(Y ) ≤ p(X ∩ Y ) + p(X ∪ Y ) whenever X and Y are crossing.
Lemma 2.4 implies that for any c : V → Z+ and any c-composition F there is a cross-free
c-composition F ′ such that
X∈F ′ p(X) ≥
X∈F p(X).
2.2 Submodular and supermodular set functions
2.2.1 Basic properties
A set function b : 2 V → Z ∪ {∞} is called fully submodular (or submodular for short) if
b(X) + b(Y ) ≤ b(X ∩ Y ) + b(X ∪ Y ) (2.1)
holds for every X, Y ⊆ V . It is clear from the definition that the sum of fully submodular set
functions is fully submodular. A modular set function (obtained from a function m : V → Z
by m(X) :=
v∈X m(v)) is always fully submodular. There is an alternative way to
characterize submodularity:
Proposition 2.6. A set function b : 2 V → Z ∪ {∞} is fully submodular if and only if
for all X ⊆ Y ⊂ V and v ∈ V − Y .
b(X + v) − b(X) ≥ b(Y + v) − b(Y )