5,333 research outputs found
Packing Steiner Trees
Let be a distinguished subset of vertices in a graph . A
-\emph{Steiner tree} is a subgraph of that is a tree and that spans .
Kriesell conjectured that contains pairwise edge-disjoint -Steiner
trees provided that every edge-cut of that separates has size .
When a -Steiner tree is a spanning tree and the conjecture is a
consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved
that Kriesell's conjecture holds when is replaced by , and recently
West and Wu have lowered this value to . Our main result makes a further
improvement to .Comment: 38 pages, 4 figure
The cavity approach for Steiner trees packing problems
The Belief Propagation approximation, or cavity method, has been recently
applied to several combinatorial optimization problems in its zero-temperature
implementation, the max-sum algorithm. In particular, recent developments to
solve the edge-disjoint paths problem and the prize-collecting Steiner tree
problem on graphs have shown remarkable results for several classes of graphs
and for benchmark instances. Here we propose a generalization of these
techniques for two variants of the Steiner trees packing problem where multiple
"interacting" trees have to be sought within a given graph. Depending on the
interaction among trees we distinguish the vertex-disjoint Steiner trees
problem, where trees cannot share nodes, from the edge-disjoint Steiner trees
problem, where edges cannot be shared by trees but nodes can be members of
multiple trees. Several practical problems of huge interest in network design
can be mapped into these two variants, for instance, the physical design of
Very Large Scale Integration (VLSI) chips. The formalism described here relies
on two components edge-variables that allows us to formulate a massage-passing
algorithm for the V-DStP and two algorithms for the E-DStP differing in the
scaling of the computational time with respect to some relevant parameters. We
will show that one of the two formalisms used for the edge-disjoint variant
allow us to map the max-sum update equations into a weighted maximum matching
problem over proper bipartite graphs. We developed a heuristic procedure based
on the max-sum equations that shows excellent performance in synthetic networks
(in particular outperforming standard multi-step greedy procedures by large
margins) and on large benchmark instances of VLSI for which the optimal
solution is known, on which the algorithm found the optimum in two cases and
the gap to optimality was never larger than 4 %
Perfect Omniscience, Perfect Secrecy and Steiner Tree Packing
We consider perfect secret key generation for a ``pairwise independent
network'' model in which every pair of terminals share a random binary string,
with the strings shared by distinct terminal pairs being mutually independent.
The terminals are then allowed to communicate interactively over a public
noiseless channel of unlimited capacity. All the terminals as well as an
eavesdropper observe this communication. The objective is to generate a perfect
secret key shared by a given set of terminals at the largest rate possible, and
concealed from the eavesdropper.
First, we show how the notion of perfect omniscience plays a central role in
characterizing perfect secret key capacity. Second, a multigraph representation
of the underlying secrecy model leads us to an efficient algorithm for perfect
secret key generation based on maximal Steiner tree packing. This algorithm
attains capacity when all the terminals seek to share a key, and, in general,
attains at least half the capacity. Third, when a single ``helper'' terminal
assists the remaining ``user'' terminals in generating a perfect secret key, we
give necessary and sufficient conditions for the optimality of the algorithm;
also, a ``weak'' helper is shown to be sufficient for optimality.Comment: accepted to the IEEE Transactions on Information Theor
Graphs with large generalized (edge-)connectivity
The generalized -connectivity of a graph , introduced by
Hager in 1985, is a nice generalization of the classical connectivity.
Recently, as a natural counterpart, we proposed the concept of generalized
-edge-connectivity . In this paper, graphs of order such
that and for even
are characterized.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1207.183
The generalized 3-edge-connectivity of lexicographic product graphs
The generalized -edge-connectivity of a graph is a
generalization of the concept of edge-connectivity. The lexicographic product
of two graphs and , denoted by , is an important graph
product. In this paper, we mainly study the generalized 3-edge-connectivity of
, and get upper and lower bounds of .
Moreover, all bounds are sharp.Comment: 14 page
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