54,998 research outputs found
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
A Superstabilizing -Approximation Algorithm for Dynamic Steiner Trees
In this paper we design and prove correct a fully dynamic distributed
algorithm for maintaining an approximate Steiner tree that connects via a
minimum-weight spanning tree a subset of nodes of a network (referred as
Steiner members or Steiner group) . Steiner trees are good candidates to
efficiently implement communication primitives such as publish/subscribe or
multicast, essential building blocks for the new emergent networks (e.g. P2P,
sensor or adhoc networks). The cost of the solution returned by our algorithm
is at most times the cost of an optimal solution, where is the
group of members. Our algorithm improves over existing solutions in several
ways. First, it tolerates the dynamism of both the group members and the
network. Next, our algorithm is self-stabilizing, that is, it copes with nodes
memory corruption. Last but not least, our algorithm is
\emph{superstabilizing}. That is, while converging to a correct configuration
(i.e., a Steiner tree) after a modification of the network, it keeps offering
the Steiner tree service during the stabilization time to all members that have
not been affected by this modification
Robust capacitated trees and networks with uniform demands
We are interested in the design of robust (or resilient) capacitated rooted
Steiner networks in case of terminals with uniform demands. Formally, we are
given a graph, capacity and cost functions on the edges, a root, a subset of
nodes called terminals, and a bound k on the number of edge failures. We first
study the problem where k = 1 and the network that we want to design must be a
tree covering the root and the terminals: we give complexity results and
propose models to optimize both the cost of the tree and the number of
terminals disconnected from the root in the worst case of an edge failure,
while respecting the capacity constraints on the edges. Second, we consider the
problem of computing a minimum-cost survivable network, i.e., a network that
covers the root and terminals even after the removal of any k edges, while
still respecting the capacity constraints on the edges. We also consider the
possibility of protecting a given number of edges. We propose three different
formulations: a cut-set based formulation, a flow based one, and a bilevel one
(with an attacker and a defender). We propose algorithms to solve each
formulation and compare their efficiency
Thresholded Covering Algorithms for Robust and Max-Min Optimization
The general problem of robust optimization is this: one of several possible
scenarios will appear tomorrow, but things are more expensive tomorrow than
they are today. What should you anticipatorily buy today, so that the
worst-case cost (summed over both days) is minimized? Feige et al. and
Khandekar et al. considered the k-robust model where the possible outcomes
tomorrow are given by all demand-subsets of size k, and gave algorithms for the
set cover problem, and the Steiner tree and facility location problems in this
model, respectively.
In this paper, we give the following simple and intuitive template for
k-robust problems: "having built some anticipatory solution, if there exists a
single demand whose augmentation cost is larger than some threshold, augment
the anticipatory solution to cover this demand as well, and repeat". In this
paper we show that this template gives us improved approximation algorithms for
k-robust Steiner tree and set cover, and the first approximation algorithms for
k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios
(except for multicut) are almost best possible.
As a by-product of our techniques, we also get algorithms for max-min
problems of the form: "given a covering problem instance, which k of the
elements are costliest to cover?".Comment: 24 page
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