Edge-connectivity of undirected and directed hypergraphs

Chapter 4
Connectivity augmentation of
directed hypergraphs
In Section 1.3 it was shown that edge-connectivity of directed hypergraphs can be described
in similar terms as edge-connectivity of digraphs. Consequently, the tools available for
solving to edge-connectivity problems are also similar; however, there are some additional
difficulties that must be overcome.
In this chapter we discuss edge-connectivity augmentation problems for directed hy-
pergraphs. As in the undirected case, there are different ways to generalize the basic
degree-specified and minimum cardinality digraphs problems. One possibility is minimum
cardinality augmentation with restriction on the sizes of the new hyperarcs. Similarly to
the undirected case, the problem with no restrictions is not interesting, since one could
always choose hyperarcs containing every node of V . Unlike the undirected case, however,
the augmentation with digraph edges is not more difficult than the digraph augmentation
problem. The other possible objective is to minimize the total size of the added hyperarcs.
The main tool used in this chapter is a slight generalization of the method developed in
[7] for splitting off in directed hypergraphs. This will be described in Section 4.2, after a
brief introduction on augmentation with digraph edges. The chapter contains joint results
with Márton Makai that appeared in [52].
4.1 Adding digraph edges
For a given directed hypergraph D0 = (V, A0), let p(X) := (k − ϱD0(X)) + for every
∅ = X ⊂ V , and p(∅) = p(V ) := 0. Then the k-edge-connectivity augmentation of D
with a minimum number of digraph edges corresponds to the problem of covering p with a
minimum number of digraph edges. The set function p is positively crossing supermodular,
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