48,038 research outputs found
Approximating the generalized terminal backup problem via half-integral multiflow relaxation
We consider a network design problem called the generalized terminal backup
problem. Whereas earlier work investigated the edge-connectivity constraints
only, we consider both edge- and node-connectivity constraints for this
problem. A major contribution of this paper is the development of a strongly
polynomial-time 4/3-approximation algorithm for the problem. Specifically, we
show that a linear programming relaxation of the problem is half-integral, and
that the half-integral optimal solution can be rounded to a 4/3-approximate
solution. We also prove that the linear programming relaxation of the problem
with the edge-connectivity constraints is equivalent to minimizing the cost of
half-integral multiflows that satisfy flow demands given from terminals. This
observation presents a strongly polynomial-time algorithm for computing a
minimum cost half-integral multiflow under flow demand constraints
Short Cycles Connectivity
Short cycles connectivity is a generalization of ordinary connectivity.
Instead by a path (sequence of edges), two vertices have to be connected by a
sequence of short cycles, in which two adjacent cycles have at least one common
vertex. If all adjacent cycles in the sequence share at least one edge, we talk
about edge short cycles connectivity.
It is shown that the short cycles connectivity is an equivalence relation on
the set of vertices, while the edge short cycles connectivity components
determine an equivalence relation on the set of edges. Efficient algorithms for
determining equivalence classes are presented.
Short cycles connectivity can be extended to directed graphs (cyclic and
transitive connectivity). For further generalization we can also consider
connectivity by small cliques or other families of graphs
Accurately modeling the Internet topology
Based on measurements of the Internet topology data, we found out that there
are two mechanisms which are necessary for the correct modeling of the Internet
topology at the Autonomous Systems (AS) level: the Interactive Growth of new
nodes and new internal links, and a nonlinear preferential attachment, where
the preference probability is described by a positive-feedback mechanism. Based
on the above mechanisms, we introduce the Positive-Feedback Preference (PFP)
model which accurately reproduces many topological properties of the AS-level
Internet, including: degree distribution, rich-club connectivity, the maximum
degree, shortest path length, short cycles, disassortative mixing and
betweenness centrality. The PFP model is a phenomenological model which
provides a novel insight into the evolutionary dynamics of real complex
networks.Comment: 20 pages and 17 figure
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