11,839 research outputs found
Approximation algorithms for Capacitated Facility Location Problem with Penalties
In this paper, we address the problem of capacitated facility location
problem with penalties (CapFLPP) paid per unit of unserved demand. In case of
uncapacitated FLP with penalties demands of a client are either entirely met or
are entirely rejected and penalty is paid. In the uncapacitated case, there is
no reason to serve a client partially. Whereas, in case of CapFLPP, it may be
beneficial to serve a client partially instead of not serving at all and, pay
the penalty for the unmet demand. Charikar et. al.
\cite{charikar2001algorithms}, Jain et. al. \cite{jain2003greedy} and Xu- Xu
\cite{xu2009improved} gave , and approximation, respectively,
for the uncapacitated case . We present factor for the case
of uniform capacities and factor for non-uniform
capacities
An Improved Approximation Algorithm for the Hard Uniform Capacitated k-median Problem
In the -median problem, given a set of locations, the goal is to select a
subset of at most centers so as to minimize the total cost of connecting
each location to its nearest center. We study the uniform hard capacitated
version of the -median problem, in which each selected center can only serve
a limited number of locations.
Inspired by the algorithm of Charikar, Guha, Tardos and Shmoys, we give a
-approximation algorithm for this problem with increasing the
capacities by a factor of , which improves
the previous best -approximation algorithm proposed by Byrka,
Fleszar, Rybicki and Spoerhase violating the capacities by factor
.Comment: 19 pages, 1 figur
Algorithms for Constructing Overlay Networks For Live Streaming
We present a polynomial time approximation algorithm for constructing an
overlay multicast network for streaming live media events over the Internet.
The class of overlay networks constructed by our algorithm include networks
used by Akamai Technologies to deliver live media events to a global audience
with high fidelity. We construct networks consisting of three stages of nodes.
The nodes in the first stage are the entry points that act as sources for the
live streams. Each source forwards each of its streams to one or more nodes in
the second stage that are called reflectors. A reflector can split an incoming
stream into multiple identical outgoing streams, which are then sent on to
nodes in the third and final stage that act as sinks and are located in edge
networks near end-users. As the packets in a stream travel from one stage to
the next, some of them may be lost. A sink combines the packets from multiple
instances of the same stream (by reordering packets and discarding duplicates)
to form a single instance of the stream with minimal loss. Our primary
contribution is an algorithm that constructs an overlay network that provably
satisfies capacity and reliability constraints to within a constant factor of
optimal, and minimizes cost to within a logarithmic factor of optimal. Further
in the common case where only the transmission costs are minimized, we show
that our algorithm produces a solution that has cost within a factor of 2 of
optimal. We also implement our algorithm and evaluate it on realistic traces
derived from Akamai's live streaming network. Our empirical results show that
our algorithm can be used to efficiently construct large-scale overlay networks
in practice with near-optimal cost
An approximation algorithm for a facility location problem with stochastic demands
In this article we propose, for any , a -approximation algorithm for a facility location problem with stochastic demands. This problem can be described as follows. There are a number of locations, where facilities may be opened and a number of demand points, where requests for items arise at random. The requests are sent to open facilities. At the open facilities, inventory is kept such that arriving requests find a zero inventory with (at most) some pre-specified probability. After constant times, the inventory is replenished to a fixed order up to level. The time interval between consecutive replenishments is called a reorder period. The problem is where to locate the facilities and how to assign the demand points to facilities at minimal cost per reorder period such that the above mentioned quality of service is insured. The incurred costs are the expected transportation costs from the demand points to the facilities, the operating costs (opening costs) of the facilities and the investment in inventory (inventory costs). \u
Centrality of Trees for Capacitated k-Center
There is a large discrepancy in our understanding of uncapacitated and
capacitated versions of network location problems. This is perhaps best
illustrated by the classical k-center problem: there is a simple tight
2-approximation algorithm for the uncapacitated version whereas the first
constant factor approximation algorithm for the general version with capacities
was only recently obtained by using an intricate rounding algorithm that
achieves an approximation guarantee in the hundreds.
Our paper aims to bridge this discrepancy. For the capacitated k-center
problem, we give a simple algorithm with a clean analysis that allows us to
prove an approximation guarantee of 9. It uses the standard LP relaxation and
comes close to settling the integrality gap (after necessary preprocessing),
which is narrowed down to either 7, 8 or 9. The algorithm proceeds by first
reducing to special tree instances, and then solves such instances optimally.
Our concept of tree instances is quite versatile, and applies to natural
variants of the capacitated k-center problem for which we also obtain improved
algorithms. Finally, we give evidence to show that more powerful preprocessing
could lead to better algorithms, by giving an approximation algorithm that
beats the integrality gap for instances where all non-zero capacities are
uniform.Comment: 21 pages, 2 figure
Investigations on two classes of covering problems
Covering problems fall within the broader category of facility location, a branch of combinatorial
optimization concerned with the optimal placement of service facilities in some
geometric space. This thesis considers two classes of covering problems. The first, Covering
with Variable Capacities (CVC), was introduced in [1] and adds a notion of capacity
to the classical Uncapacitated Facility Location problem. That is, each facility has a fixed
maximum quantity of clients it can serve. The objective of each variant of CVC is either to
serve all clients, the greatest number of clients possible, or all clients using the least number
of facilities possible. We provide approximation algorithms, and in a few select cases,
optimal algorithms, for all three variants of CVC.
The second class of covering problems is barrier coverage. When the purpose of coverage
is surveillance rather than service, a cost effective approach to the problem of intruder
detection is to place sensors along the boundary, or barrier, of the surveilled region. A
barrier coverage is complete when any intrusion is sure to be detected by some sensor. We
limit our consideration of barrier coverage to the one-dimensional case, where the region is
a line segment. Sensors are themselves line segments, whose span forms a detection range.
The objective of barrier coverage as considered here is to form a complete barrier coverage
while minimizing the total movement cost, the sum of the weighted distances moved by
each sensor in the solution. We show that, by assuming the sensors lie in initial positions
where their detection ranges are disjoint from the barrier, one-dimensional barrier coverage
can be solved with an FPTAS. Along the way to developing the FPTAS, we give a fast,
simple 2-approximation algorithm for weighted disjoint barrier coverage
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