5,167 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
Speeding-up Dynamic Programming with Representative Sets - An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions
Dynamic programming on tree decompositions is a frequently used approach to
solve otherwise intractable problems on instances of small treewidth. In recent
work by Bodlaender et al., it was shown that for many connectivity problems,
there exist algorithms that use time, linear in the number of vertices, and
single exponential in the width of the tree decomposition that is used. The
central idea is that it suffices to compute representative sets, and these can
be computed efficiently with help of Gaussian elimination.
In this paper, we give an experimental evaluation of this technique for the
Steiner Tree problem. A comparison of the classic dynamic programming algorithm
and the improved dynamic programming algorithm that employs the table reduction
shows that the new approach gives significant improvements on the running time
of the algorithm and the size of the tables computed by the dynamic programming
algorithm, and thus that the rank based approach from Bodlaender et al. does
not only give significant theoretical improvements but also is a viable
approach in a practical setting, and showcases the potential of exploiting the
idea of representative sets for speeding up dynamic programming algorithms
The Fast Heuristic Algorithms and Post-Processing Techniques to Design Large and Low-Cost Communication Networks
It is challenging to design large and low-cost communication networks. In
this paper, we formulate this challenge as the prize-collecting Steiner Tree
Problem (PCSTP). The objective is to minimize the costs of transmission routes
and the disconnected monetary or informational profits. Initially, we note that
the PCSTP is MAX SNP-hard. Then, we propose some post-processing techniques to
improve suboptimal solutions to PCSTP. Based on these techniques, we propose
two fast heuristic algorithms: the first one is a quasilinear time heuristic
algorithm that is faster and consumes less memory than other algorithms; and
the second one is an improvement of a stateof-the-art polynomial time heuristic
algorithm that can find high-quality solutions at a speed that is only inferior
to the first one. We demonstrate the competitiveness of our heuristic
algorithms by comparing them with the state-of-the-art ones on the largest
existing benchmark instances (169 800 vertices and 338 551 edges). Moreover, we
generate new instances that are even larger (1 000 000 vertices and 10 000 000
edges) to further demonstrate their advantages in large networks. The
state-ofthe-art algorithms are too slow to find high-quality solutions for
instances of this size, whereas our new heuristic algorithms can do this in
around 6 to 45s on a personal computer. Ultimately, we apply our
post-processing techniques to update the bestknown solution for a notoriously
difficult benchmark instance to show that they can improve near-optimal
solutions to PCSTP. In conclusion, we demonstrate the usefulness of our
heuristic algorithms and post-processing techniques for designing large and
low-cost communication networks
Vertex and edge covers with clustering properties: complexity and algorithms
We consider the concepts of a t-total vertex cover and a t-total edge cover (t≥1), which generalise the notions of a vertex cover and an edge cover, respectively. A t-total vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has at least t vertices (edges). These definitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t, we present NP-completeness and approximability results (both upper and lower bounds) and FTP algorithms for problems concerned with finding the minimum size of a t-total vertex cover, t-total edge cover and connected vertex cover, in particular improving on a previous FTP algorithm for the latter problem
Discrete Particle Swarm Optimization for the minimum labelling Steiner tree problem
Particle Swarm Optimization is an evolutionary method inspired by the
social behaviour of individuals inside swarms in nature. Solutions of the problem are
modelled as members of the swarm which fly in the solution space. The evolution is
obtained from the continuous movement of the particles that constitute the swarm
submitted to the effect of the inertia and the attraction of the members who lead the
swarm. This work focuses on a recent Discrete Particle Swarm Optimization for combinatorial optimization, called Jumping Particle Swarm Optimization. Its effectiveness is
illustrated on the minimum labelling Steiner tree problem: given an undirected labelled
connected graph, the aim is to find a spanning tree covering a given subset of nodes,
whose edges have the smallest number of distinct labels
Recommended from our members
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
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