7,156 research outputs found
Covering problems in edge- and node-weighted graphs
This paper discusses the graph covering problem in which a set of edges in an
edge- and node-weighted graph is chosen to satisfy some covering constraints
while minimizing the sum of the weights. In this problem, because of the large
integrality gap of a natural linear programming (LP) relaxation, LP rounding
algorithms based on the relaxation yield poor performance. Here we propose a
stronger LP relaxation for the graph covering problem. The proposed relaxation
is applied to designing primal-dual algorithms for two fundamental graph
covering problems: the prize-collecting edge dominating set problem and the
multicut problem in trees. Our algorithms are an exact polynomial-time
algorithm for the former problem, and a 2-approximation algorithm for the
latter problem, respectively. These results match the currently known best
results for purely edge-weighted graphs.Comment: To appear in SWAT 201
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
Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices
We study the Steiner Tree problem, in which a set of terminal vertices needs
to be connected in the cheapest possible way in an edge-weighted graph. This
problem has been extensively studied from the viewpoint of approximation and
also parametrization. In particular, on one hand Steiner Tree is known to be
APX-hard, and W[2]-hard on the other, if parameterized by the number of
non-terminals (Steiner vertices) in the optimum solution. In contrast to this
we give an efficient parameterized approximation scheme (EPAS), which
circumvents both hardness results. Moreover, our methods imply the existence of
a polynomial size approximate kernelization scheme (PSAKS) for the considered
parameter.
We further study the parameterized approximability of other variants of
Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of
these an EPAS is likely to exist for the studied parameter: for Steiner Forest
an easy observation shows that the problem is APX-hard, even if the input graph
contains no Steiner vertices. For Directed Steiner Tree we prove that
approximating within any function of the studied parameter is W[1]-hard.
Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree,
but a PSAKS does not. We also prove that there is an EPAS and a PSAKS for
Steiner Forest if in addition to the number of Steiner vertices, the number of
connected components of an optimal solution is considered to be a parameter.Comment: 23 pages, 6 figures An extended abstract appeared in proceedings of
STACS 201
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