Edge-connectivity of undirected and directed hypergraphs

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70 Chapter 4. Connectivity augmentation of directed hypergraphs so the following theorem of Frank [24] can be applied to solve the degree-constrained problem: Theorem 4.1 (Frank [24]). Let p : 2 V → Z+ be a positively crossing supermodular set function; let mi : V → Z+ be an indegree-specification and mo : V → Z + an outdegree- specification such that mi(V ) = mo(V ). There exists a digraph D = (V, A) such that ϱD(v) = mi(v) and δD(v) = mo(v) ∀v ∈ V , and ϱD(X) ≥ p(X) ∀X ⊆ V if and only if and mi(X) ≥ p(X) for every X ⊆ V , mo(V − X) ≥ p(X) for every X ⊆ V . It should be noted that for the p defined above this result can be obtained from the directed splitting-off theorem of Mader [59], which is generalized by the above result of Frank. The degree-specified result easily implies the following theorem on minimum cardinality augmentation: Theorem 4.2 (Frank [24]). Let p : 2 V → Z+ be a positively crossing supermodular set function, and γ a non-negative integer. There exists a digraph D = (V, A) with γ edges that covers p if and only if and γ ≥ p(X) for every partition F of V , Z∈F γ ≥ p(V − X) for every partition F of V . Z∈F The minimum cost k-edge-connectivity augmentation problem for digraphs is NP-complete even for k = 1, since it includes the problem of determining whether a digraph has a Hamiltonian cycle. For undirected graphs, the local edge-connectivity augmentation problem was solvable [29] using Theorem 3.3 of Mader on splitting-off preserving local edge-connectivity. How- ever, no such result can be expected for digraphs: local edge-connectivity augmentation of directed graphs is NP-complete, even in the special case when the requirement is 1 between the nodes of a specified subset T and 0 otherwise (see [29]).
Section 4.2. Splitting-off in directed hypergraphs 71 4.2 Splitting-off in directed hypergraphs In [7], Berg, Jackson and Jordán proved an interesting splitting-off theorem for directed hypergraphs, which led to a solution for the problem of directed hypergraph k-edge- connectivity augmentation by uniform hyperarcs. In this section, we show that their edge-splitting result can be formulated in a more general form (using essentially the same proof). The result gives a method for solving a broader class of undirected and directed augmentation problems. The notion of splitting-off is used here in an abstract sense, similarly to Theorem 4.5 or Theorem 3.14. 4.2.1 Splitting off a single hyperarc Let p : 2 V → Z+ be a positively crossing supermodular set function. Let furthermore mi : V → Z+ be an indegree-specification and mo : V → Z+ an outdegree-specification for which mi(V ) ≤ mo(V ). Suppose that mi(X) ≥ p(X) and mo(V − X) ≥ p(X) for every X ⊆ V . We define the splitting-off operation analogously to the undirected definition in the proof of Theorem 3.14. A hyperarc a can be split off from (p, mi, mo) if mi(h(a)) > 0 and χt(a)(v) ≤ mo(v) for every v ∈ V . For such a hyperarc let m a ⎧ ⎨mi(v) − 1 if v = h(a), i (v) := ⎩mi(v) otherwise, m a o(v) :=mo(v) − χt(a)(v), p a ⎧ ⎨(p(X) − 1) (X) := ⎩ + if a enters X, p(X) otherwise. The splitting-off operation is feasible if m a i (X) ≥ p a (X) and m a o(V − X) ≥ p a (X) for every X ⊆ V . The operation is called a feasible (r, 1)-splitting if |a| = r + 1. It is easy to check using (1.4) that p a is positively crossing supermodular. To motivate the name “splitting-off”, we may observe that if we add a new node z to V , and draw mi(v) digraph edges from z to each node v ∈ V , and mo(v) edges from each node v ∈ V to z, then the above defined splitting-off operation corresponds to splitting- off in this digraph D. If in addition p is defined as p(X) = (k − ϱD0(X)) + for some digraph D0, a feasible splitting-off corresponds to a splitting-off in D0 + D that preserves k-edge-connectivity in V (see Figure 4.1). The following theorem describes conditions when a feasible splitting-off is available.
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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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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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