Edge-connectivity of undirected and directed hypergraphs

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74 Chapter 4. Connectivity augmentation of directed hypergraphs hypergraph D and positive integer k, p(X) = (k − ϱD(X)) + for every ∅ = X ⊂ V ) was proved in [7]. The more general result presented here has the advantage that it can be applied to extensions of k-edge-connectivity as well (while the proof is not more difficult). Theorem 4.5. Let p : 2 V → Z+ be a positively crossing supermodular set function, mi : V → Z+ and mo : V → Z+ degree specifications such that mo(V ) = rmi(V ) for some positive integer r, and mi(X) ≥ p(X) for every X ⊆ V , (4.3) mo(V − X) ≥ p(X) for every X ⊆ V . (4.4) Then there is a directed (r, 1)-hypergraph D such that δD(v) = mo(v) and ϱD(v) = mi(v) for every v ∈ V , and ϱD(X) ≥ p(X) for every X ⊆ V . Proof. According to Theorem 4.3 we can obtain a directed hypergraph D ∗ by successive feasible splitting-off operations such that δD ∗(v) = mo(v), ϱD ∗(v) = mi(v) for every v ∈ V , and ϱD ∗(X) ≥ p(X) for every X ⊆ V . Since mo(V ) = rmi(V ), |a| ≥ r + 1 holds for at least one hyperarc a of D ∗ . So there is a feasible (r1, 1)-splitting with head h(a) for some r1 ≥ r; moreover, by Theorem 4.3 there is also a feasible (r2, 1)-splitting with head h(a) for some r2 ≤ r. Lemma 4.6. If for some r1 > r > r2 there is a feasible (r1, 1)-spitting and a feasible (r2, 1)-splitting with head u , then there is a feasible (r, 1)-splitting with head u. Proof. Let a be the hyperarc obtained by the (r1, 1)-splitting. By induction, it suffices to show that for some v ∈ t(a), the hyperarc a ′ defined by a ′ = a−v, h(a ′ ) = u gives a feasible splitting. If χa(v) ≥ 2 for some v ∈ V then such a v is suitable, so we may assume that χa ≤ 1. If r1 > 2, then suppose indirectly that for every v ∈ t(a) there is an in-critical set Xv such that a − v ⊆ Xv and v /∈ Xv. We can assume that these are maximal in-critical sets. Thus the sets {Xv : v ∈ t(a)} form a flower with r1 petals centered on u; but this contradicts the fact that there is a feasible (r2, 1)-splitting with head u. If r1 = 2, then r2 = 0, so there are no in-critical sets. As we have seen in the proof of Theorem 4.3, there is a unique minimal out-critical set Y with u /∈ Y . Then a − Y = u, otherwise the (2, 1)-splitting would not be feasible; thus both (1, 1)-splittings are feasible. We prove Theorem 4.5 by induction on mi(V ). According to Lemma 4.6, there is a node u ∈ V with mi(u) > 0 for which there exists a feasible (r, 1)-splitting at u; let a be
Section 4.3. Covering supermodular functions by directed hypergraphs 75 the resulting (r, 1)-hyperarc. By induction, there is a directed (r, 1)-hypergraph D ′ that satisfies the conditions given by m a i , m a o and p a . The directed hypergraph obtained by adding a to D ′ satisfies the conditions of Theorem 4.5. In fact, a bit more general theorem can also be proved by the same method (this was also remarked by Berg, Jackson, and Jordán for the k-edge-connectivity case): Theorem 4.7. Let p : 2 V → Z+ be a positively crossing supermodular set function, mi : V → Z+ and mo : V → Z+ degree specifications such that mi(V ) ≤ mo(V ) and (r − 1)mi(V ) < mo(V ) ≤ rmi(V ) for some positive integer r, and mi(X) ≥ p(X) for every X ⊆ V , (4.5) mo(V − X) ≥ p(X) for every X ⊆ V . (4.6) Then there is a directed hypergraph D consisting of (r, 1)-hyperedges and (r − 1, 1)-hyperedges such that δD(v) = mo(v) and ϱD(v) = mi(v) for every v ∈ V , and ϱD(X) ≥ p(X) for every X ⊆ V . Proof. By Theorem 4.3 there is a complete splitting-off, so there must be an (r1, 1)-hyperarc a that can be feasibly split off for some r1 ≥ r, since (r − 1)mi(V ) < mo(V ). By Theorem 4.3 there is also an (r2, 1)-hyperarc for some r2 ≤ r with head h(a) that can be feasibly split off, since mo(V ) ≤ rmi(V ). So Lemma 4.6 implies that there is an (r, 1)-hyperarc that can be feasibly split off. We can continue to split off (r, 1)-hyperarcs as long as (r − 1)m ′ i(V ) < m ′ o(V ) ≤ rm ′ i(V ) holds for the modified degree-specifications. It is easy to see that the first time this does not hold is when (r − 1)m ′ i(V ) = m ′ o(V ). But then by Theorem 4.5 we can finish the complete splitting-off by splitting off (r−1, 1)-hyperarcs. 4.3 Covering supermodular functions by directed hypergraphs Let p : 2 V → Z+ be a positively crossing supermodular set function. When covering p with a directed hypergraph, we can have various objectives just as in the case of undirected hypergraphs: we can minimize the total size of the added hyperarcs, or we can cover p with an (r, 1)-uniform directed hypergraph of minimum cardinality.
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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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Page 24 and 25: 24 Chapter 1. Introduction and prel
Page 26 and 27: 26 Chapter 1. Introduction and prel
Page 28 and 29: 28 Chapter 2. Submodular functions
Page 48 and 49: 48 Chapter 3. Edge-connectivity aug
Page 70 and 71: 70 Chapter 4. Connectivity augmenta
Page 84 and 85: 84 Chapter 5. Partition-connectivit
Page 96 and 97: 96 Chapter 6. Hypergraph orientatio
Page 98 and 99: 98 Chapter 6. Hypergraph orientatio
Page 100 and 101: 100 Chapter 6. Hypergraph orientati
Page 116 and 117: 116 Chapter 7. Combined augmentatio
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124 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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