241 research outputs found
LP-Based Algorithms for Capacitated Facility Location
Linear programming has played a key role in the study of algorithms for
combinatorial optimization problems. In the field of approximation algorithms,
this is well illustrated by the uncapacitated facility location problem. A
variety of algorithmic methodologies, such as LP-rounding and primal-dual
method, have been applied to and evolved from algorithms for this problem.
Unfortunately, this collection of powerful algorithmic techniques had not yet
been applicable to the more general capacitated facility location problem. In
fact, all of the known algorithms with good performance guarantees were based
on a single technique, local search, and no linear programming relaxation was
known to efficiently approximate the problem.
In this paper, we present a linear programming relaxation with constant
integrality gap for capacitated facility location. We demonstrate that the
fundamental theories of multi-commodity flows and matchings provide key
insights that lead to the strong relaxation. Our algorithmic proof of
integrality gap is obtained by finally accessing the rich toolbox of LP-based
methodologies: we present a constant factor approximation algorithm based on
LP-rounding.Comment: 25 pages, 6 figures; minor revision
An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem
We obtain a 1.5-approximation algorithm for the metric uncapacitated facility
location problem (UFL), which improves on the previously best known
1.52-approximation algorithm by Mahdian, Ye and Zhang. Note, that the
approximability lower bound by Guha and Khuller is 1.463.
An algorithm is a {\em (,)-approximation algorithm} if
the solution it produces has total cost at most , where and are the facility and the connection
cost of an optimal solution. Our new algorithm, which is a modification of the
-approximation algorithm of Chudak and Shmoys, is a
(1.6774,1.3738)-approximation algorithm for the UFL problem and is the first
one that touches the approximability limit curve
established by Jain, Mahdian and Saberi. As a consequence, we obtain the first
optimal approximation algorithm for instances dominated by connection costs.
When combined with a (1.11,1.7764)-approximation algorithm proposed by Jain et
al., and later analyzed by Mahdian et al., we obtain the overall approximation
guarantee of 1.5 for the metric UFL problem. We also describe how to use our
algorithm to improve the approximation ratio for the 3-level version of UFL.Comment: A journal versio
Constant-Factor FPT Approximation for Capacitated k-Median
Capacitated k-median is one of the few outstanding optimization problems for which the existence of a polynomial time constant factor approximation algorithm remains an open problem. In a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities k, the problem is also W[2] hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time 2^O(k log k) n^O(1) and achieves an approximation ratio of 7+epsilon
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
Online Facility Location with Deletions
In this paper we study three previously unstudied variants of the online Facility Location problem, considering an intrinsic scenario when the clients and facilities are not only allowed to arrive to the system, but they can also depart at any moment.
We begin with the study of a natural fully-dynamic online uncapacitated model where clients can be both added and removed. When a client arrives, then it has to be assigned either to an existing facility or to a new facility opened at the client\u27s location. However, when a client who has been also one of the open facilities is to be removed, then our model has to allow to reconnect all clients that have been connected to that removed facility. In this model, we present an optimal O(log(n_{act}) / log log(n_{act}))-competitive algorithm, where n_{act} is the number of active clients at the end of the input sequence.
Next, we turn our attention to the capacitated Facility Location problem. We first note that if no deletions are allowed, then one can achieve an optimal competitive ratio of O(log(n) / log(log n)), where n is the length of the sequence. However, when deletions are allowed, the capacitated version of the problem is significantly more challenging than the uncapacitated one. We show that still, using a more sophisticated algorithmic approach, one can obtain an online O(log N + log c log n)-competitive algorithm for the capacitated Facility Location problem in the fully dynamic model, where N is number of points in the input metric and c is the capacity of any open facility
Facility Location in Evolving Metrics
Understanding the dynamics of evolving social or infrastructure networks is a
challenge in applied areas such as epidemiology, viral marketing, or urban
planning. During the past decade, data has been collected on such networks but
has yet to be fully analyzed. We propose to use information on the dynamics of
the data to find stable partitions of the network into groups. For that
purpose, we introduce a time-dependent, dynamic version of the facility
location problem, that includes a switching cost when a client's assignment
changes from one facility to another. This might provide a better
representation of an evolving network, emphasizing the abrupt change of
relationships between subjects rather than the continuous evolution of the
underlying network. We show that in realistic examples this model yields indeed
better fitting solutions than optimizing every snapshot independently. We
present an -approximation algorithm and a matching hardness result,
where is the number of clients and the number of time steps. We also
give an other algorithms with approximation ratio for the variant
where one pays at each time step (leasing) for each open facility
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