Edge-connectivity of undirected and directed hypergraphs

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116 Chapter 7. Combined augmentation and orientation izations become much more complicated in that case, and the solvability of minimum cardinality augmentation remains open; we only address the degree-specified problem. The new results presented here are partly from [35], a joint paper with András Frank, and partly from [52], a joint work with Márton Makai. It should be noted that the theorems of the chapter give new results even in the special case when they are restricted to graphs. For example, a min-max theorem on minimum cardinality (k, l)-partition-connectivity augmentation of graphs can be derived from Corollary 7.11. 7.2 Augmentation to meet orientability requirements 7.2.1 Degree specified augmentation As in the case of undirected and directed edge-connectivity augmentation, we first prove a degree specified augmentation result. The conditions on the requirement function are a bit more restrictive now, since we require non-negativity, like in Theorem 6.11. Note that Theorem 6.11 actually corresponds to the special case of the following theorem when the degree specification is 0 on every node. We have seen that there can be different objectives in an augmentation problem, depend- ing on the restrictions on the sizes of the added hyperedges. The results presented here give characterizations for both the uniform and the unrestricted augmentation problem. The given characterizations are “good” in the sense that they provide an easily verifiable certificate if the augmentation is impossible. Theorem 7.1. Let H0 = (V, E0) be a hypergraph, p : 2 V → Z+ a non-negative crossing supermodular set function, m : V → Z+ a degree specification, and 0 ≤ γ ≤ m(V )/2 an integer. There exists a hypergraph H = (V, E) with γ hyperedges such that H0 + H has an orientation covering p and dH(v) = m(v) for every v ∈ V if and only if the following hold for every partition F of V : γ ≥ p(Z) − eH0(F), (7.1) Z∈F min F ′ ⊆F, X=∪F ′ (m(V − X) + (|F ′ | − 1)γ) ≥ In addition, H can be chosen so that m(V ) m(V ) ≤ |e| ≤ γ γ min m(V − X) ≥ p(Z) − eH0(F), (7.2) X∈F Z∈F Z∈F p(V − Z) − eH0(co(F)). (7.3) for every e ∈ E. (7.4)
Section 7.2. Augmentation to meet orientability requirements 117 Proof. The right hand side of the inequalities is the deficiency of the hyperedges of H0 on the appropriate families. The necessity of the conditions follows from the observation that the left hand side is always an upper bound on the contribution of the new hyperarcs. In (7.1): every new hyperarc can enter at most one set of F; in (7.2): every hyperarc that enters a set of F must have a node in V − X; in (7.3): the number of sets of co(F) that a new hyperarc enters is at most |F ′ | − 1 plus the number of nodes it has in V − X. The proof of sufficiency is based on the orientation of an extended hypergraph. We add a new node z to the set of nodes, and for every v ∈ V we add m(v) parallel edges between v and z; the resulting hypergraph is denoted by H ′ 0 = (V ′ , E ′ 0). Our first aim is to find an orientation H ′ 0 of H ′ 0 that has the following properties: ϱH ′ (V ) = γ, (7.5) 0 ϱH ′ (X) ≥ p(X) if ∅ = X ⊂ V , (7.6) 0 ϱH ′ (X + z) ≥ p(X) if ∅ = X ⊂ V . (7.7) 0 To find such an orientation, we use Lemma 6.10. A vector x : V → Z+ is called feasible if it is the vector of in-degrees (restricted to V ) of an orientation satisfying (7.5)–(7.7). It is easy to see using Lemma 6.10 that x is feasible if and only if x(V ) = |E0| + γ and x(Z) ≥ pm(Z) for every Z ⊆ V , where pm(X) := p(X) + iH0(X) + (γ − m(V − X)) + (X ⊆ V ). (7.8) Thus a vector x is feasible if and only if it is an integral element of B(pm) (as defined in (2.5)). Claim 7.2. The set function pm is crossing supermodular. Proof. The crossing supermodularity of p and (1.3) implies that p + iH0 is crossing supermodular. Let m ∗ (X) := (γ−m(V −X)) + ; we show that this set function is fully supermodular. If m ∗ (Y ) = 0, then m ∗ (X)+m ∗ (Y ) = m ∗ (X) ≤ m ∗ (X∪Y ) = m ∗ (X∩Y )+m ∗ (X∪Y ). If m ∗ (X), m ∗ (Y ) > 0, then m ∗ (X) + m ∗ (Y ) = 2γ − 2m(V ) + m(X ∩ Y ) + m(X ∪ Y ) ≤ m ∗ (X ∩ Y ) + m ∗ (X ∪ Y ). The sum of a crossing supermodular and a fully supermodular function is crossing supermodular. Theorem 2.11 implies that if B(pm) is non-empty, then it is a base polyhedron, hence its vertices are integral. Claim 7.3. If conditions (7.1)-(7.3) are satisfied, then B(pm) is non-empty.
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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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28 Chapter 2. Submodular functions
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48 Chapter 3. Edge-connectivity aug
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Page 66 and 67: 66 Chapter 3. Edge-connectivity aug
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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 118 and 119: 118 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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