Edge-connectivity of undirected and directed hypergraphs

8 Chapter 1. Introduction and preliminaries
tween undirected and directed graphs) to hypergraphs, and on the other hand, to present
a framework for combining connectivity augmentation and orientation problems. Both
of these themes are addressed with a strong emphasis on the role of submodularity and
related properties intrinsic to the structure of these problems.
The understanding of the links to submodularity provides an important tool for obtaining
structural descriptions and min-max theorems for the different variants of edge-connectivity
properties for graphs, hypergraphs and directed hypergraphs. Moreover, the well-known
algorithmic properties of submodular systems imply that efficient algorithms can be con-
structed based on the structural results. The precise analysis and optimization of these
algorithms is beyond the scope of the thesis, usually only the existence of polynomial time
algorithms will be stated.
There is a lot of ongoing research in the field, most of which is not addressed here in
detail (for example the very interesting class of problems featuring both connectivity and
parity), so the thesis is by no means a comprehensive survey. Rather, it intends to provide
a concise account of connections and structural relations concerning a well-defined class
of problems. This includes several new results on hypergraph edge-connectivity, some of
which are interesting even when specialized to graphs.
1.1.2 Edge-connectivity – brief history
The most basic properties of edge-connectivity were established by different versions of
Menger’s classical theorem on the relation between cuts and edge-disjoint paths, and by
the concept of network flows [21]. Important structural results were obtained by Gomory
and Hu [42] and Edmonds [16]. Another significant contribution to the development of the
field was the study of packings and coverings by trees, where the fundamental results are
due to Tutte [70] and Nash-Williams [63]. Analogous theorems for directed graphs were
established by Edmonds [14].
These areas are closely linked to the theory of matroids and submodular functions.
We do not give a detailed account of the many fundamental results on submodularity by
Edmonds, Lovász and others; for a concise bibliography see [26]. Efficient frameworks
for dealing with connectivity problems in the context of submodularity were given for
example in [28] and [66]. One notion of particular importance is that of submodular flows,
introduced by Edmonds and Giles [17], which is the most popular general algorithmic
framework featuring submodular functions.
One major development of the research going on in the field was the successful work
on edge-connectivity augmentation problems. Initial results were obtained by Eswaran
and Tarjan [18], and Plesnik [64]. Fundamental deep results were due to Lovász [55]