47,646 research outputs found
Approximation algorithms for facility location problems
Facility location problems are among the most well-studied problems in optimization literature. The simplest variant is the uncapacitated facility location problem, which we denoted by the UFLP. In the UFLP, we are given a set of possible facility locations and a set of clients. The problem seeks to find a set of locations to build/open facilities such that the sum of the cost of building/opening the facilities and the cost of serving each client from exactly one open facility is minimized. This problem is NP−hard. Therefore, many heuristics for finding good approximate solutions were developed. In this thesis we design approximation algorithms for several variants of the UFLP
Large-Scale Distributed Algorithms for Facility Location with Outliers
This paper presents fast, distributed, O(1)-approximation algorithms for metric facility location problems with outliers in the Congested Clique model, Massively Parallel Computation (MPC) model, and in the k-machine model. The paper considers Robust Facility Location and Facility Location with Penalties, two versions of the facility location problem with outliers proposed by Charikar et al. (SODA 2001). The paper also considers two alternatives for specifying the input: the input metric can be provided explicitly (as an n x n matrix distributed among the machines) or implicitly as the shortest path metric of a given edge-weighted graph. The results in the paper are:
- Implicit metric: For both problems, O(1)-approximation algorithms running in O(poly(log n)) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model.
- Explicit metric: For both problems, O(1)-approximation algorithms running in O(log log log n) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model.
Our main contribution is to show the existence of Mettu-Plaxton-style O(1)-approximation algorithms for both Facility Location with outlier problems. As shown in our previous work (Berns et al., ICALP 2012, Bandyapadhyay et al., ICDCN 2018) Mettu-Plaxton style algorithms are more easily amenable to being implemented efficiently in distributed and large-scale models of computation
Approximation Algorithms for Continuous Clustering and Facility Location Problems
We consider the approximability of center-based clustering problems where the
points to be clustered lie in a metric space, and no candidate centers are
specified. We call such problems "continuous", to distinguish from "discrete"
clustering where candidate centers are specified. For many objectives, one can
reduce the continuous case to the discrete case, and use an
-approximation algorithm for the discrete case to get a
-approximation for the continuous case, where depends on
the objective: e.g. for -median, , and for -means, . Our motivating question is whether this gap of is inherent, or are
there better algorithms for continuous clustering than simply reducing to the
discrete case? In a recent SODA 2021 paper, Cohen-Addad, Karthik, and Lee prove
a factor- and a factor- hardness, respectively, for continuous -median
and -means, even when the number of centers is a constant. The discrete
case for a constant is exactly solvable in polytime, so the loss
seems unavoidable in some regimes.
In this paper, we approach continuous clustering via the round-or-cut
framework. For four continuous clustering problems, we outperform the reduction
to the discrete case. Notably, for the problem -UFL, where
and the discrete case has a hardness of , we obtain an approximation
ratio of for the continuous case. Also, for continuous
-means, where the best known approximation ratio for the discrete case is
, we obtain an approximation ratio of . The key challenge
is that most algorithms for discrete clustering, including the state of the
art, depend on linear programs that become infinite-sized in the continuous
case. To overcome this, we design new linear programs for the continuous case
which are amenable to the round-or-cut framework.Comment: 24 pages, 0 figures. Full version of ESA 2022 paper
https://drops.dagstuhl.de/opus/volltexte/2022/16971 . This version adds a
link to the conference version and fixes minor formatting issue
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
Approximation algorithms for facility location problems. In:
Abstract We present new approximation algorithms for several facility location problems. In each facility location problem that we study, there is a set of locations at which w e m a y build a facility such a s a w arehouse, where the cost of building at location i is fi; furthermore, there is a set of client locations such a s stores that require to be serviced by a facility, and if a client at location j is assigned to a facility at location i, a cost of cij is incurred. The objective i s t o determine a set of locations at which to open facilities so as to minimize the total facility and assignment costs. In the uncapacitated case, each facility can service an unlimited number of clients, whereas in the capacitated case, each facility can serve, for example, at most u clients. These models and a number of closely related ones have been studied extensively in the Operations Research literature. We shall consider the case in which the assignment costs are symmetric and satisfy the triangle inequality. For the uncapacitated facility location, we give a polynomial-time algorithm that nds a solution within a factor of 3.16 of the optimal. This is the rst constant performance guarantee known for this problem. We also present approximation algorithms with constant performance guarantees for a number of capacitated models as well as a generalization in which there is a 2-level hierarchy of facilities. Our results are based on the ltering and rounding technique of Lin & Vitter. We also give a randomized variant of this technique that can then be derandomized to yield improved performance guarantees
A new approximation algorithm for the multilevel facility location problem
In this paper we propose a new integer programming formulation for the multi-level facility location problem and a novel 3-approximation algorithm based on LP rounding. The linear program we are using has a polynomial number of variables and constraints, being thus more efficient than the one commonly used in the approximation algorithms for this type of problems
Probabilistic Analysis of Facility Location on Random Shortest Path Metrics
The facility location problem is an NP-hard optimization problem. Therefore,
approximation algorithms are often used to solve large instances. Such
algorithms often perform much better than worst-case analysis suggests.
Therefore, probabilistic analysis is a widely used tool to analyze such
algorithms. Most research on probabilistic analysis of NP-hard optimization
problems involving metric spaces, such as the facility location problem, has
been focused on Euclidean instances, and also instances with independent
(random) edge lengths, which are non-metric, have been researched. We would
like to extend this knowledge to other, more general, metrics.
We investigate the facility location problem using random shortest path
metrics. We analyze some probabilistic properties for a simple greedy heuristic
which gives a solution to the facility location problem: opening the
cheapest facilities (with only depending on the facility opening
costs). If the facility opening costs are such that is not too large,
then we show that this heuristic is asymptotically optimal. On the other hand,
for large values of , the analysis becomes more difficult, and we
provide a closed-form expression as upper bound for the expected approximation
ratio. In the special case where all facility opening costs are equal this
closed-form expression reduces to or or even
if the opening costs are sufficiently small.Comment: A preliminary version accepted to CiE 201
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