Edge-connectivity of undirected and directed hypergraphs

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68 Chapter 3. Edge-connectivity augmentation of hypergraphs To solve the minimum cardinality problem in polynomial time we have to find a minimal degree specification m that satisfies (3.9). This can be done in polynomial time using the maximizing oracle. In the case of k-edge-connectivity augmentation, the required oracles can be realized using network flow algorithms.
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 hypergraphs. 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, 69
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Edge-connectivity of undirected and
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4 Contents 6 Hypergraph orientation
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6 Contents
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8 Chapter 1. Introduction and preli
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10 Chapter 1. Introduction and prel
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118 Chapter 7. Combined augmentatio
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130 Chapter 8. Concluding remarks w
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132 Bibliography [11] E. Cheng, Edg
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134 Bibliography [38] S. Fujishige,
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136 Bibliography [66] A. Schrijver,
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138 Index in-degree of node, 19 ind
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140 Index h(a) the head node of the
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142 Index
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Edge-connectivity of undirected and directed hypergraphs

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