7,420 research outputs found
A Solution Merging Heuristic for the Steiner Problem in Graphs Using Tree Decompositions
Fixed parameter tractable algorithms for bounded treewidth are known to exist
for a wide class of graph optimization problems. While most research in this
area has been focused on exact algorithms, it is hard to find decompositions of
treewidth sufficiently small to make these al- gorithms fast enough for
practical use. Consequently, tree decomposition based algorithms have limited
applicability to large scale optimization. However, by first reducing the input
graph so that a small width tree decomposition can be found, we can harness the
power of tree decomposi- tion based techniques in a heuristic algorithm, usable
on graphs of much larger treewidth than would be tractable to solve exactly. We
propose a solution merging heuristic to the Steiner Tree Problem that applies
this idea. Standard local search heuristics provide a natural way to generate
subgraphs with lower treewidth than the original instance, and subse- quently
we extract an improved solution by solving the instance induced by this
subgraph. As such the fixed parameter tractable algorithm be- comes an
efficient tool for our solution merging heuristic. For a large class of sparse
benchmark instances the algorithm is able to find small width tree
decompositions on the union of generated solutions. Subsequently it can often
improve on the generated solutions fast
Variational approximation of functionals defined on 1-dimensional connected sets: the planar case
In this paper we consider variational problems involving 1-dimensional
connected sets in the Euclidean plane, such as the classical Steiner tree
problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal
partition problems and provide a variational approximation through
Modica-Mortola type energies proving a -convergence result. We also
introduce a suitable convex relaxation and develop the corresponding numerical
implementations. The proposed methods are quite general and the results we
obtain can be extended to -dimensional Euclidean space or to more general
manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure
Variable neighbourhood search for the minimum labelling Steiner tree problem
We present a study on heuristic solution approaches to the minimum labelling Steiner
tree problem, an NP-hard graph problem related to the minimum labelling spanning tree
problem. Given an undirected labelled connected graph, the aim is to find a spanning
tree covering a given subset of nodes of the graph, whose edges have the smallest number
of distinct labels. Such a model may be used to represent many real world problems in
telecommunications and multimodal transportation networks. Several metaheuristics are
proposed and evaluated. The approaches are compared to the widely adopted Pilot Method
and it is shown that the Variable Neighbourhood Search metaheuristic is the most effective
approach to the problem, obtaining high quality solutions in short computational running
times
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Variable neighbourhood search for the minimum labelling Steiner tree problem
We present a study on heuristic solution approaches to the minimum labelling Steiner tree problem, an NP-hard graph problem related to the minimum labelling spanning tree problem. Given an undirected labelled connected graph, the aim is to find a spanning tree covering a given subset of nodes of the graph, whose edges have the smallest number of distinct labels. Such a model may be used to represent many real world problems in telecommunications and multimodal transportation networks. Several metaheuristics are proposed and evaluated. The approaches are compared to the widely adopted Pilot Method and it is shown that the Variable Neighbourhood Search that we propose is the most effective metaheuristic for the problem, obtaining high quality solutions in short computational running time
Concurrent Geometric Multicasting
We present MCFR, a multicasting concurrent face routing algorithm that uses
geometric routing to deliver a message from source to multiple targets. We
describe the algorithm's operation, prove it correct, estimate its performance
bounds and evaluate its performance using simulation. Our estimate shows that
MCFR is the first geometric multicast routing algorithm whose message delivery
latency is independent of network size and only proportional to the distance
between the source and the targets. Our simulation indicates that MCFR has
significantly better reliability than existing algorithms
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