Edge-connectivity of undirected and directed hypergraphs

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