Edge-connectivity of undirected and directed hypergraphs

Chapter 7
Combined augmentation and
orientation
7.1 Introduction
The problems discussed in this chapter have ties to both connectivity orientation and
connectivity augmentation. The results of Chapter 6 suggest that the property that a
hypergraph has an orientation with high edge-connectivity may itself be considered as a
connectivity property. For example, the existence of a (k, l)-edge-connected orientation
is equivalent to the (k, l)-partition-connectivity of the hypergraph (defined in Chapter
5). It is reasonable to ask when can connectivity augmentation be solved for this kind
of connectivity properties. In other words, we examine the solvability of the following
problem: given a hypergraph H0, add an optimal set of hyperedges to H0 such that the
resulting hypergraph has an orientation with some prescribed connectivity property.
At first glance this question could seem to be closely related to the undirected augmen-
tation problems described in Chapter 3. Interestingly, it turns out that it has more in
common with the connectivity augmentation of directed hypergraphs, discussed in Chap-
ter 4. In particular, we will extensively use the directed splitting-off technique described
there.
Problems will be presented in the usual framework involving the covering of set func-
tions, but in the first sections we restrict ourselves to non-negative crossing supermodular
set functions. This class includes the requirement function for (k, l)-edge-connected ori-
entations, and the obtained results give min-max formulas on (k, l)-partition-connectivity
augmentation.
The final section contains some results on combined augmentation-orientation problems
when the requirement function is only positively crossing supermodular. The character-
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