Edge-connectivity of undirected and directed hypergraphs

98 Chapter 6. Hypergraph orientation
x1
x1
zc1
sc1
tc1
wc1
xi1
xi1
zci
xi2
xi2
sci
tci
xi3
xi3
zck
xl
xl
tck
wci sck wck
Figure 6.1: Reduction of 3-SAT to local edge-connectivity orientation. The figure shows
the construction for the clause ci = (xi1, xi2, xi3). An orientation of the edges of type xx
corresponds to an evaluation; the orientation of the other edges is uniquely determined.
vyzc. Consider the problem of finding an orientation of G such that for every clause c ∈ C
there are at least 3 edge-disjoint paths from sc to wc, 3 edge-disjoint paths from zc to tc,
and 1 path from sc to tc. It is easy to see that the existence of such an orientation is
equivalent to the satisfiability of C.
6.2.2 Orientations and submodular flows
The orientation problems described above can be studied for mixed graphs (graphs with
both undirected and directed edges) as well. An orientation of a mixed graph is obtained by
orienting its undirected edges. In [25] Frank solved the problem of finding an orientation of
a mixed graph that covers a given crossing supermodular set function which does not have
to be non-negative. He showed that this is equivalent to a submodular flow problem, and
as a result minimum cost orientation (when the two possible orientations of an undirected
edge have different costs) can be solved in polynomial time. The characterization here is
considerably more complicated then in Theorem 6.3:
Theorem 6.5 (Frank [25]). Let G = (V ; E, A) be a mixed graph, where E is the set of
undirected edges, and A is the set of directed edges. Let p : 2 V → Z ∪ {−∞} be a crossing
supermodular set function. Then G has an orientation covering p if and only if
(p(Z) − ϱA(Z)) ≤ eE(F)
Z∈F
holds whenever F is a tree-composition of some X ⊆ V .
The reason that tree-compositions come into the picture is the connection with submod-
ular flows: in Theorem 2.28 we saw that the condition on the feasibility of a submodular
sci
tci