Edge-connectivity of undirected and directed hypergraphs

Section 6.5. Local connectivity requirements 109
corresponding undirected edge in E. The capacities of the edges are bounded by f ≡ 0,
g ≡ 1.
For a {0, 1}-valued cost function c, consider the system (S), and let the dual solutions
be denoted by (y, z), where y consists of the dual variables associated to the constraints
in (6.8). Take an integral dual optimal solution (y ∗ , z ∗ ) ≥ 0, where y ∗ is positive on a
laminar family, and |z ∗ | is minimal. Let F be the laminar family where every set X has
multiplicity y ∗ (X). It follows from the minimality of |z ∗ | that the objective value of this
dual solution is
p(X) −
(ee(F) − c(e)) + −
ρa(X) − c(a)
X∈F
e∈E
a∈A
X∈F
+
=
=
p(X) − eE(F) −
ϱA(X) + µ(F).
Conversely, the value of (6.14) corresponds to the value of the following dual solution
(y ∗ , z ∗ ). Let F be a laminar family where the maximum is attained in (6.14). For X ⊆ V ,
let y ∗ (X) be the multiplicity of X in F. Define the values of the dual variables in z ∗ as
required by the dual constraints, always setting a variable corresponding to an edge to 0
if the edge also belongs to an orientation constraint. In this case the dual objective value
is equal to the expression (6.14).
X∈F
6.5 Local connectivity requirements
Theorems 6.13 and 6.14 show that in relation to the covering of intersecting supermodular
set functions, a relatively large class of orientation problems can be efficiently solved. The
preceding results of the chapter indicate that a somewhat more restricted class is solvable
concerning the covering of crossing supermodular set functions. In this section we attempt
to further relax the condition of crossing supermodularity. As we have mentioned at the
beginning of the chapter, it is open whether some analogue of Theorem 6.4 can be proved
for hypergraphs. The problems discussed in this section are much less ambitious, but
nevertheless they have some interesting corollaries.
6.5.1 Local requirement for one pair of nodes
We consider k-edge-connected orientations of graphs and hypergraphs, where the number
of edge-disjoint paths required between two designated special nodes may be more than
k. First we formulate a partition-type condition for the hypergraph case, and prove its
sufficiency using a modified uncrossing method.
X∈F