Edge-connectivity of undirected and directed hypergraphs

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44 Chapter 2. Submodular functions Proposition 2.25. Let N = N(W, A1, A2) be a network matrix. Let l : A1 → Z ∪ {−∞}, u : A1 → Z ∪ {∞} be lower and upper capacities on A1 such that l ≤ u, and let f : A2 → Z ∪ {−∞}, g : A2 → Z ∪ {∞} be lower and upper capacities on A2 such that f ≤ g. Then the system is TDI. {x : A2 → Q : l ≤ Nx ≤ u, f ≤ x ≤ g} (2.25) To see an example, consider a digraph D = (V, A), and a cross-free family F on the ground set V . Let (T = (W, A1), ϕ) be the tree-representation of F. We can associate a digraph D ′ = (W, A2) to D by taking the edge ϕ(u)ϕ(v) for every edge uv ∈ A. We get a network matrix N = N(W, A1, A2). The system (2.25) described in Proposition 2.25 corresponds to the following system, if we consider l, u to be given on the sets of F, and f, g to be given on the edges of D: {x : A → Q : l(Z) ≤ δx(Z) − ϱx(Z) ≤ u(Z) ∀Z ∈ F, f ≤ x ≤ g}, (2.26) where δx(Z) = a∈∆ + D (Z) x(a) and ϱx(Z) = system is TDI. 2.4.2 Submodular flows a∈∆ − D (Z) x(a). So by proposition 2.25, this To simplify the notations later in the thesis, we describe submodular flows using supermodular functions, but this is of course equivalent to the submodular formulation. Let D = (V, A) be a digraph, and p : 2 V → Z∪{−∞} a crossing supermodular function. Let furthermore f : A → Z ∪ {−∞}, g : A → Z ∪ {∞} be upper and lower capacities on A such that f ≤ g. For x : A → Q, let φx(Z) := δx(Z) − ϱx(Z), which is a modular function. The system {x : A → Q : φx(Z) ≥ p(Z) ∀Z ⊆ V, f ≤ x ≤ g} (2.27) is called a submodular flow system. A simple example is system (2.26), where p(X) = max{l(X), −u(V − X)} if X ∈ F + co(F) ,and p(X) = −∞ otherwise. A submodular flow system is one-way if p(X) > −∞ implies that either ϱD(X) = 0 or δD(X) = 0. It is strongly one-way if either ϱD(X) = 0 whenever p(X) > −∞, or δD(X) = 0 whenever p(X) > −∞. The fundamental result on submodular flows is the following: Theorem 2.26 (Edmonds and Giles[17]). The system (2.27) is TDI.
Section 2.5. Algorithms 45 Algorithms for finding a minimum cost submodular flow will be mentioned in Section 2.5. The solvability of system (2.27) has a simple characterization if p is fully supermodular: Theorem 2.27 (Frank [23]). Suppose that p is fully supermodular. Then (2.27) has a solution if and only if δg(Z) − ϱf(Z) ≥ p(Z) for every Z ⊆ V . If there is a solution, then there is an integral solution as well. Suppose now that p is crossing supermodular, and p(V ) = 0 (this can be assumed since (2.27) has no solution if p(V ) > 0). Since φx is modular, the system (2.27) can be written in the form {x : A → Q : φx ∈ B(p), f ≤ x ≤ g}. (2.28) If B(p) is non-empty, then by Proposition 2.15, the set function p ↑ is fully supermodular, and B(p) = B(p ↑ ). Thus Theorem 2.27 implies the following: Theorem 2.28 (Frank [23]). Suppose that p is crossing supermodular. Then (2.27) has a solution if and only if δg(Z) − ϱf(Z) ≥ p ↑ (Z) for every Z ⊆ V , where p ↑ is the full truncation of p, defined in (2.11). If there is a solution, then there is an integral solution as well. 2.5 Algorithms 2.5.1 Oracles When set functions are considered from an algorithmic point of view, the way of getting in- formation about their values must be clarified. Many set functions in the thesis are defined using a graph or a hypergraph, and in these cases the values usually can be computed using network flows. However, in the general case, we consider the values to be given by some kind of oracle. We distinguish two kinds of oracles for a set function b : 2 V → Z ∪ {∞}: Evaluation oracle: Provides the value of b(X) for any subset X ⊆ V . It also tells for s, t ∈ V whether there is an st-set Z with b(Z) < ∞. Minimizing oracle: Given ∅ ⊆ X ⊆ Y ⊆ V and a modular function m, it calculates min{b(Z) − m(Z) : X ⊆ Z ⊆ Y }. A maximizing oracle for a set function p : 2 V → Z ∪ {−∞} is a minimizing oracle for −p. It was shown in [32] that the upper truncation 2.10 of an intersecting supermodular
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Page 4 and 5: 4 Contents 6 Hypergraph orientation
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94 Chapter 5. Partition-connectivit
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96 Chapter 6. Hypergraph orientatio
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98 Chapter 6. Hypergraph orientatio
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100 Chapter 6. Hypergraph orientati
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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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