Edge-connectivity of undirected and directed hypergraphs

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50 Chapter 3. Edge-connectivity augmentation of hypergraphs 3.1.3 Hypergraphs: various objectives Edge-connectivity augmentation problems for hypergraphs have been less intensively studied than their graph counterparts. However, some recent results have shown that they are also worthy of interest. In [11], Cheng gave a formula on the minimum number of graph edges that can be added to an initial (k − 1)-edge-connected hypergraph so that the resulting hypergraph is k-edge-connected; Bang-Jensen and Jackson [2] extended this result to the case when the initial hypergraph can be arbitrary. More general frameworks, involving the covering of set functions or of families of sets, were proposed by Benczúr and Frank [9] and Fleiner and Jordán [20]. Szigeti [69] solved the local edge-connectivity augmentation problem for hypergraphs when the aim is to minimize the total size of the added hyperedges. These results show that the objective of minimizing the number of new edges, which we used in the graph case, can be generalized in various directions. In the rest of the chapter we address these possibilities. Of course minimizing the number of new hyperedges is in itself an uninteresting question, since it is always best to use hyperedges containing the whole ground set. One alternative is to consider instead the total size (the sum of the sizes) of the hyperedges; another is to constrain the size of the added hyperedges. In the following sections we cite the known results about these objectives. We also describe the generalization studied in [9] and [69], that involves covering certain types of set functions by a hypergraph. The chapter is concluded by a new result on covering symmetric crossing supermodular functions by uniform hypergraphs, which extends results in [20] and [9]. 3.2 Minimum size augmentation 3.2.1 Local edge-connectivity requirements Let us first consider the objective of minimizing the sum of the sizes of the added hyperedges, called the total size. It turns out that in some sense this problem is actually easier then graph edge-connectivity augmentation. Local edge-connectivity augmentation can be solved, and it does not require tools as sophisticated as Theorem 3.3 of Mader. Let r : V 2 → Z+ be a local edge-connectivity requirement function for which r(u, v) = r(v, u) and r(v, v) = 0 for every u, v ∈ V . For a set X ⊆ V , let R(X) := max{r(u, v) : u ∈ X, v ∈ V − X}. Szigeti [69] proved the following: Theorem 3.8 (Szigeti [69]). Let H0 = (V, E0) be a hypergraph, and r a local edge- connectivity requirement function like above. There is a hypergraph H with hyperedges of
Section 3.3. Augmentation using graph edges 51 total size σ such that λH0+H(u, v) ≥ r(u, v) for every u, v ∈ V if and only if Z∈F (R(Z) − dH0(Z)) ≤ σ holds for every subpartition F of V . The proof also yields a polynomial-time algorithm for the solution. 3.2.2 Covering skew supermodular functions In the above problem, finding an augmenting hypergraph with minimum total size amounts to finding a hypergraph of minimum total size that covers the set function R(Z) − dH0(Z). As it was mentioned in Section 2.3, this set function is symmetric and skew supermodular. So a natural generalization of the augmentation problem is to find a hypergraph with minimum total size that covers a given symmetric skew supermodular set function. Actually it was this problem that was solved by Szigeti: Theorem 3.9 (Szigeti [69]). Let p : 2 V → Z be a symmetric skew-supermodular set function. There is a hypergraph H with hyperedges of total size σ that covers p if and only if σ ≥ Z∈F p(Z) for every subpartition F of V . Note that the symmetry of p is not crucial: if p is skew supermodular, then the set function p ′ (X) := max{p(X), p(V − X)} is symmetric and skew supermodular. 3.3 Augmentation using graph edges An alternative to minimum total size is to consider minimum cardinality augmentation, but with a constraint on the size of the added hyperedges. The most simple case is when we allow only the addition of graph edges. 3.3.1 NP-completeness of local edge-connectivity augmentation As we indicated, there is a polynomial algorithm for local edge-connectivity augmentation of hypergraphs with a minimum total size of hyperedges. It was also mentioned that using Theorem 3.3, local edge-connectivity augmentation of graphs using a minimum number of edges is solvable. Though this raised hopes that the minimum cardinality problem for hypergraphs, i.e the minimization of the number of new graph edges added to the initial hypergraph, might also be tractable for local edge-connectivity augmentation, it turned out that this problem is NP-complete. NP-completeness was proved by Cosh, Jackson, and Z. Király for the following special case:
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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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Edge-connectivity of undirected and directed hypergraphs

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