Attila Bernáth

A new approach to splitting-off

A new approach to undirected splitting-off is presented in this paper. We study the behaviour of splitting-off algorithms when applied to the problem of covering a symmetric skew-supermodular set function by a graph. This hard problem is a natural generalization of many solved connectivity augmentation problems, such as local edgeconnectivity augmentation of graphs, global arc-connectivity augmentation of mixed graphs with undirected edges, or the node-to-area connectivity augmentation problem in graphs. Using a simple lemma we characterize the situation when a splitting-off algorithm can be stuck. This characterization enables us to give very simple proofs for the classical results mentioned above. Finally we apply our observations in generalizations of the above problems: we consider two connectivity augmentation problems in hypergraphs. The first is the local edgeconnectivity augmentation of undirected hypergraphs by hyperedges of minimum total size without increasing the rank. The second is global arc-connectivity augmentation of mixed hypergraphs by adding hyperedges of minimum total size without increasing the rank. We show that a greedy approach works in (almost) all of these cases.

The Generalized Terminal Backup Problem

We consider the following network design problem that we call the Generalized Terminal Backup Problem: given a graph (or a hypergraph)

Blocking optimal k -arborescences

G_0=(V, {E}_0)

Augmenting the edge-connectivity of a hypergraph by adding a multipartite graph

, a set of (at least 2) terminals

Well-balanced orientations of mixed graphs

T\subseteq V

Source location in undirected and directed hypergraphs

, and a requirement

Augmenting the Edge‐Connectivity of a Hypergraph by Adding a Multipartite Graph

r(t)

Worst case bin packing for OTN electrical layer networks dimensioning

for every

The complexity of the Clar number problem and an FPT algorithm.

t\in T

Partition Constrained Covering of a Symmetric Crossing Supermodular Function by a Graph

, find a multigraph