Edge-connectivity of undirected and directed hypergraphs

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66 Chapter 3. Edge-connectivity augmentation of hypergraphs We prove sufficiency by showing that if a p-full partition F = {V1, . . . , Vl} violates (3.23) while conditions (3.24) and (3.25) are satisfied, then an appropriate E ′ 0 violates (3.26). Let E ′ 0 be the set of hyperedges in E0 that enter at least one member of F, and let H ′ 0 := (V, E ′ 0). Then dH ′ (U) ≤ k − 1 if U is the union of some (not all) members of F, since F is p-full. 0 The partition F violates (3.23), so (ν − 1)γ < l − 1 ≤ c(H0 − E ′ 0) − 1. We claim that |E ′ 0| = k − 1, which then implies that (3.26) is violated. By (3.24), lk − l i=1 dH ′ 0 (Vi) = lk − l i=1 dH0(Vi) ≤ νγ ≤ ν ν−1 (l − 2) ≤ 2l − 4, from which l i=1 dH ′ 0 (Vi) ≥ (k − 2)l + 4. This implies that |E ′ 0| ≥ k − 1, and there are at least 4 members of F (say V1, V2, V3, V4), for which dH ′ 0 (Vi) = k − 1. We can assume that iH ′ 0 (V1 ∪ V2) ≤ iH ′ 0 (Vi ∪ Vj) for every distinct i, j ∈ {1, 2, 3, 4}. If iH ′ 0 (V1∪V2) > 0, then dH ′ 0 (V1∪V2) ≥ dH ′ 0 (V1)−iH ′ 0 (V1∪V2)+iH ′ 0 (V2∪V3)+iH ′ 0 (V2∪V4) ≥ k, contradicting the p-fullness of F. So iH ′ 0 (V1 ∪ V2) = 0, in which case there are k − 1 hyperedges in E ′ 0 that enter each of V1, V2, and V1 ∪ V2. Suppose that E ′ 0 contains a hyperedge besides these k − 1, which enters a partition member Vi. Then dH ′ 0 (V1 ∪ Vi) ≥ k, which would contradict the p-fullness of F. This proves that |E ′ 0| = k − 1. 3.4.5 Algorithmic aspects It might be argued that Theorems 3.14 and 3.29 are not good characterizations, since it is not possible to check in polynomial time whether a given partition is p-full, hence it cannot be decided whether it is a deficient partition or not. Indeed, it is well known that it is NP-complete to decide for given graph G = (V, E) whether there is a set X ⊆ V with dG(X) ≥ k. Let F be the partition of V composed of singleton members, and let p(X) := (k − dG(X)) + for ∅ = X ⊂ V and p(∅) = p(V ) := 0. Now F is a p-full partition if and only if dG(X) < k for every X ⊆ V . If a partition has at least one member Vi with p(Vi) = 1, then its deficiency can be checked using the characterization in Claim 3.17. But in general, deciding whether a partition is p-full or not is NP-complete. However, it is easy to see that if p(Vi) ≥ 2 for every member of a deficient partition, then at least one of the partition members violates (3.9) (and the partition violates (3.21) in case of Theorem 3.29). This means that Theorems 3.14 and 3.29 give good co-NP characterizations. The proof of Theorem 3.14 provides a polynomial algorithm for the degree specified problem if maximizing oracles are available for every set function of the form p − dH, where H is an arbitrary hypergraph. In this case, a feasible splitting-off operation can be found in polynomial time using the procedure described in the proof. However, we have to introduce a kind of “multiple splitting-off” to ensure that the number of splitting-off steps is polynomial.
Section 3.4. Covering by uniform hypergraphs 67 We choose a hyperedge e that can be feasibly split off by the method described in the proof of Theorem 3.14. If m(v) ≥ ν for every v ∈ e then the following extended splitting-off operation may be used: (⋆) We determine the maximal µ for which e can be feasibly split off µ times and µχe(v) ≤ m(v) − ν + 1 for every v ∈ e. Claim 3.31. An extended splitting-off can be executed in polynomial time. Proof. The degree restriction ensures that condition (3.17) is indifferent when calculating the number of feasible splitting-off operations, since the auxiliary graph is the same before each splitting-off. So it suffices to determine the maximal value µ for which the following hold: µχe(v) ≤ m(v) − (ν − 1) for every v ∈ e, (3.27) µ(|e ∩ X| − 1) ≤ m(X) − p(X) for every X entered by e, (3.28) µν ≤ m(X) − p(X) if e ⊆ X, (3.29) µ ≤ m(V ) ν − p(X) if e ⊆ X. (3.30) The maximal µ for which (3.29) and (3.30) hold can be determined by the maximizing oracle. As for (3.28), it suffices to check its validity on the family {X ⊆ V : p(Y ) < p(X) ∀Y ⊂ X, e enters X}. This family can be determined in polynomial time since it is laminar. The extended splitting-off operation is not used if m(v) ≤ ν − 1 for some v ∈ e; a single splitting-off is executed instead. But the number such splitting-off operations is polynomial (we can assume that ν ≤ |V |). We have to prove that the number of extended splitting-off operations is also polynomial. First, observe that no set is deleted from B1 during an extended splitting-off. Indeed, X ∈ B1 is deleted during a single splitting-off of e if there is Y ⊂ X for which p(Y ) < p(X) and p e (Y ) = p e (X); but then |e ∩ (X − Y )| > 0, so m e (X − Y ) ≥ ν − 1, implying that m e (Y ) − p e (Y ) ≤ m e (X) − (ν − 1) − p e (X) < 0, which is impossible. Since B1 is always laminar, this implies that in a sequence of consecutive extended splitting-off operations, B1 can change only polynomially many times. It is easy to see that B3 also can change only polynomially many times. So it is enough to prove that if B1 and B3 do not change, then only polynomially many consecutive extended splittings are possible. But this follows from the fact that ν − 1 nodes are the same in the hyperedges that are split off.
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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 100 and 101: 100 Chapter 6. Hypergraph orientati
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116 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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