Edge-connectivity of undirected and directed hypergraphs

32 Chapter 2. Submodular functions
The rank function of the matroid M, denoted by rM, is the set function whose value on X
is the cardinality of a maximal independent subset of X. This rank function is monotone
increasing (i.e. rM(X) ≤ rM(Y ) if X ⊆ Y ), fully submodular, and rM(X) ≤ |X| for every
X ⊆ S. It can be shown that every set function satisfying these properties is the rank
function of a matroid.
Given a graph G = (V, E), One can define a matroid MG = (S, I) by S := E and
I := {E ′ ⊆ E : E ′ is a forest}. MG is called the circuit matroid of G. A generalization of
this construction to hypergraphs, due to Lorea [54], will be presented in Chapter 5.
A graph G = (V, E) is connected if rMG (E) = |V |−1. G contains k edge-disjoint spanning
trees if MG has k disjoint independent sets of cardinality |V | − 1. As the following theorem
of Edmonds asserts, this is also a matroid problem.
Theorem 2.7 (Edmonds [13]). Let Mi = (S, Ii) be matroids on a common ground set
S for i = 1, . . . , k. Then the family IΣ := {I1 ∪ I2 ∪ · · · ∪ Ik : Ii ∈ Ii} forms the family of
independent sets of a matroid MΣ whose rank-function rΣ is given by the following formula:
k
rΣ(X) = min ri(X ′ ) + |X − X ′ | : X ′
⊆ X . (2.3)
i=1
The matroid MΣ defined in the theorem is called the sum of matroids M1, . . . , Mk.
Theorem 1.5 of Tutte can be easily obtained from this result.
Another result of Edmonds that is of central importance is the matroid intersection
theorem:
Theorem 2.8 (Edmonds [15]). Let M1 = (S, I1) and M2 = (S, I2) be two matroids,
with rank functions r1 and r2. Then the maximum cardinality of a set in I1 ∩ I2 equals
min
X⊆S (r1(X) + r2(S − X)).
2.2.3 Polyhedra associated to set functions
The properties of sub- and supermodular functions can be best described using the termi-
nology of polyhedral combinatorics. Here we formulate the results from the perspective of
supermodular functions, since these appear more often in the thesis; analogous statements
are of course true for submodular functions. In the following paragraphs we use the terms
“min” and “max” to denote −∞/ + ∞ if the values are not bounded. Given a set function
p : 2 V → Z ∪ {−∞}, we define the polyhedra
C(p) := {x : V → Q : x(Y ) ≥ p(Y ) ∀Y ⊆ V } , (2.4)
B(p) := {x : V → Q : x(V ) = p(V ); x(Y ) ≥ p(Y ) ∀Y ⊆ V } . (2.5)