9,475 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
Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems
We consider geometric instances of the Maximum Weighted Matching Problem
(MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000
vertices. Making use of a geometric duality relationship between MWMP, MTSP,
and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields
in near-linear time solutions as well as upper bounds. Using various
computational tools, we get solutions within considerably less than 1% of the
optimum.
An interesting feature of our approach is that, even though an FWP is hard to
compute in theory and Edmonds' algorithm for maximum weighted matching yields a
polynomial solution for the MWMP, the practical behavior is just the opposite,
and we can solve the FWP with high accuracy in order to find a good heuristic
solution for the MWMP.Comment: 20 pages, 14 figures, Latex, to appear in Journal of Experimental
Algorithms, 200
Scalable and Cost Efficient Algorithms for Virtual CDN Migration
Virtual Content Delivery Network (vCDN) migration is necessary to optimize
the use of resources and improve the performance of the overall SDN/NFV-based
CDN function in terms of network operator cost reduction and high streaming
quality. It requires intelligent and enticed joint SDN/NFV migration algorithms
due to the evident huge amount of traffic to be delivered to end customers of
the network. In this paper, two approaches for finding the optimal and near
optimal path placement(s) and vCDN migration(s) are proposed (OPAC and HPAC).
Moreover, several scenarios are considered to quantify the OPAC and HPAC
behaviors and to compare their efficiency in terms of migration cost, migration
time, vCDN replication number, and other cost factors. Then, they are
implemented and evaluated under different network scales. Finally, the proposed
algorithms are integrated in an SDN/NFV framework. Index Terms: vCDN; SDN/NFV
Optimization; Migration Algorithms; Scalability Algorithms.Comment: 9 pages, 11 figures, 4 tableaux, conference Local Computer Networks
(LCN), class
Approximation Algorithms for Distributionally Robust Stochastic Optimization with Black-Box Distributions
Two-stage stochastic optimization is a framework for modeling uncertainty,
where we have a probability distribution over possible realizations of the
data, called scenarios, and decisions are taken in two stages: we make
first-stage decisions knowing only the underlying distribution and before a
scenario is realized, and may take additional second-stage recourse actions
after a scenario is realized. The goal is typically to minimize the total
expected cost. A criticism of this model is that the underlying probability
distribution is itself often imprecise! To address this, a versatile approach
that has been proposed is the {\em distributionally robust 2-stage model}:
given a collection of probability distributions, our goal now is to minimize
the maximum expected total cost with respect to a distribution in this
collection.
We provide a framework for designing approximation algorithms in such
settings when the collection is a ball around a central distribution and the
central distribution is accessed {\em only via a sampling black box}.
We first show that one can utilize the {\em sample average approximation}
(SAA) method to reduce the problem to the case where the central distribution
has {\em polynomial-size} support. We then show how to approximately solve a
fractional relaxation of the SAA (i.e., polynomial-scenario
central-distribution) problem. By complementing this via LP-rounding algorithms
that provide {\em local} (i.e., per-scenario) approximation guarantees, we
obtain the {\em first} approximation algorithms for the distributionally robust
versions of a variety of discrete-optimization problems including set cover,
vertex cover, edge cover, facility location, and Steiner tree, with guarantees
that are, except for set cover, within -factors of the guarantees known
for the deterministic version of the problem
The Minimum Wiener Connector
The Wiener index of a graph is the sum of all pairwise shortest-path
distances between its vertices. In this paper we study the novel problem of
finding a minimum Wiener connector: given a connected graph and a set
of query vertices, find a subgraph of that connects all
query vertices and has minimum Wiener index.
We show that The Minimum Wiener Connector admits a polynomial-time (albeit
impractical) exact algorithm for the special case where the number of query
vertices is bounded. We show that in general the problem is NP-hard, and has no
PTAS unless . Our main contribution is a
constant-factor approximation algorithm running in time
.
A thorough experimentation on a large variety of real-world graphs confirms
that our method returns smaller and denser solutions than other methods, and
does so by adding to the query set a small number of important vertices
(i.e., vertices with high centrality).Comment: Published in Proceedings of the 2015 ACM SIGMOD International
Conference on Management of Dat
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