Edge-connectivity of undirected and directed hypergraphs

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110 Chapter 6. Hypergraph orientation Theorem 6.16. Let H = (V, E) be a hypergraph, s, t ∈ V designated nodes, and k1, k2 ≥ k positive integers. H has a k-edge-connected orientation such that there are k1 edge-disjoint paths from s to t and k2 edge-disjoint paths from t to s if and only if eH(F) ≥ p(X) (6.15) X∈F for every partition F, where ⎧ 0 if X = ∅ or X = V , ⎪⎨ k1 if X is an st-set, p(X) := k2 if X is a ts-set, ⎪⎩ k otherwise. (6.16) Proof. We may assume that every hyperedge of H contains at least 2 nodes. The goal is to find an orientation of H that covers p. Observe that the set function p has none of the properties discussed in the previous sections (it is not crossing supermodular, monotone decreasing, or symmetric). As in the proof of Theorem 6.11, we increase the value of p on the singletons so that every singleton {v} is in a tight partition Fv (a partition that satisfies (6.15) by equality); let F := v∈V Fv be the sum of these partitions, and let p ′ denote the modified set function; then p ′ (X) = eH(F). (6.17) X∈F Apply one of the following three operations on F as long as any of them can be applied (see Figure 6.4): (i) Uncross X ∈ F and Y ∈ F (see Lemma 2.4) if they are crossing unless one of them is an st-set and the other is a ts-set; (ii) If F contains a co-partition, replace it by the partition obtained by taking the complement of every member; (iii) If X ∈ F is an st-set, Y ∈ F is a ts-set, and there is a sub-family G ⊆ F such that co(G) is a partition of X ∩ Y , replace X, Y and G in F by X − Y , Y − X and co(G). Claim 6.17. These operations do not increase eH(F), and do not decrease X∈F p′ (X). Proof. A simple case analysis shows that the operations do not increase eH(F), as it suffices to check that the operations do not increase X∈F ϱa(X) for any hyperarc a. An even more simple case analysis shows that the operations do not decrease X∈F p(X), consequently they cannot decrease the value X∈F p′ (X), since singletons are never removed from the family.
Section 6.5. Local connectivity requirements 111 X Y (i) (ii) (iii) Figure 6.4: The 3 operations in the proof of Theorem 6.16. The non-rounded rectangles represent their complement. Obviously F remains regular throughout the process. The second and the third operations decrease h∅(F), and by Claim 2.5 the first operation can be applied only finitely many times consecutively, so after a finite number of steps none of the three operations can be applied. Let us denote the obtained regular family by F ′ ; (6.17) and Claim 6.17 imply that X∈F ′ p ′ (X) ≥ eH(F ′ ). Let F ′′ be the regular family obtained from F ′ by decreasing the multiplicity of every singleton by one. Claim 6.18. F ′′ decomposes into partitions. Proof. We can assume that there is an st-set and a ts-set, otherwise the unavailability of the first and the second operation would imply that F ′′ is a cross-free family that decomposes into partitions. The st-sets in F ′′ form a chain, the ts sets likewise. Let X be the minimal st-set, and Y the maximal ts-set in F ′′ . If X ∩ Y = ∅, then for every v ∈ X ∩ Y there is a vt-set in F ′′ , since F ′′ is regular; let B denote the family of these sets. By the minimality of X, the members of B are not st-sets. Furthermore, they are neither crossing each other, nor X, nor Y , since the first operation cannot be applied. This is only possible if the minimal sets in co(B) define a partition of X ∩Y . But then the third operation would have been applicable, contradicting the assumption. Thus X and Y are disjoint. For every v ∈ V − X − Y there is a tv-set in F ′′ , since F ′′ is regular. By the maximality of Y , these sets are not ts-sets, so they are disjoint from X and Y , otherwise they would cross X or Y . The minimal such sets are also disjoint from each other, so they form a partition P of V − X − Y . Thus F ′′ contains the partition P + {X, Y }. By induction, F ′′ decomposes into partitions. If the conditions of the theorem are met, then X∈F ′′ p ′ (X) ≤ eH(F ′′ ) by Claim 6.18, so X∈F ′ p ′ (X) ≥ eH(F ′ ) implies that the partition formed by the singletons must be tight. Let the vector x : V → Z be defined by x(v) := p ′ ({v}); then x(V ) = |E|. t s X Y
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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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28 Chapter 2. Submodular functions
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48 Chapter 3. Edge-connectivity aug
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Page 84 and 85: 84 Chapter 5. Partition-connectivit
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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
Page 130 and 131: 130 Chapter 8. Concluding remarks w
Page 132 and 133: 132 Bibliography [11] E. Cheng, Edg
Page 134 and 135: 134 Bibliography [38] S. Fujishige,
Page 136 and 137: 136 Bibliography [66] A. Schrijver,
Page 138 and 139: 138 Index in-degree of node, 19 ind
Page 140 and 141: 140 Index h(a) the head node of the
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Edge-connectivity of undirected and directed hypergraphs

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