15,557 research outputs found
Approximation Algorithms for Clustering and Facility Location Problems
Facility location problems arise in a wide range of applications such as plant or warehouse location problems, cache placement problems, and network design problems, and have been widely studied in Computer Science and Operations Research literature. These problems typically involve an underlying set F of facilities that provide service, and an underlying set D of clients that require service, which need to be assigned to facilities in a cost-effective fashion. This abstraction is quite versatile and also captures clustering problems, where one typically seeks to partition a set of data points into k clusters, for some given k, in a suitable way, which themselves find applications in data mining, machine learning, and bioinformatics.
Basic variants of facility location problems are now relatively well-u
nderstood, but we have much-less understanding of more-sophisticated models that better model the real-world concerns. In this thesis, we focus on three models inspired by some real-world optimization scenarios.
In Chapter 2, we consider mobile facility location (MFL) problem, wherein we seek to relocate a given set of facilities to destinations closer to the clients as to minimize the sum of facility-movement and client-assignment costs. This abstracts facility-location settings where one has the flexibility of moving
facilities from their current locations to other destinations so as to serve clients more efficiently by reducing their assignment costs. We give the first local-search based approximation algorithm for this problem and
achieve the best-known approximation guarantee. Our main result is
(3+epsilon)-approximation for this problem for any constant epsilon > 0 using local
search which improves the previous best guarantee of 8-approximation algorithm due to [34] based on LP-rounding. Our results extend to the weighted generalization wherein each facility i has a
non-negative weight w_i and the movement cost for i is w_i times the distance
traveled by i.
In Chapter 3, we consider a facility-location problem that we call the minimum-load k-facility location (MLkFL), which abstracts settings where the cost of
serving the clients assigned to a facility is incurred by the facility. This problem was studied under the name of min-max star cover in [32,10], who
(among other results) gave bicriteria approximation algorithms for MLkFL when F=D. MLkFL is rather poorly understood, and only an O(k)-approximation is currently
known for MLkFL, even for line metrics. Our main result is the first polytime approximation scheme (PTAS) for MLkFL on line
metrics (note that no non-trivial true approximation of any kind was known for this metric).
Complementing this, we prove that MLkFL is strongly NP-hard on line metrics.
In Chapter 4, we consider clustering problems with non-uniform lower bounds and outliers, and
obtain the first approximation guarantees for these problems.
We consider objective functions involving the radii of open facilities, where the radius of a facility i is the maximum distance between i and a client assigned to it. We consider two problems: minimizing the sum of the radii of the open facilities, which yields the lower-bounded min-sum-of-radii with outliers (LBkSRO) problem, and minimizing the maximum radius, which yields the lower-bounded k-supplier with outliers (LBkSupO) problem. We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the non-outlier version. These also constitute the first approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers separately. We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds
Coordination of Mobile Mules via Facility Location Strategies
In this paper, we study the problem of wireless sensor network (WSN)
maintenance using mobile entities called mules. The mules are deployed in the
area of the WSN in such a way that would minimize the time it takes them to
reach a failed sensor and fix it. The mules must constantly optimize their
collective deployment to account for occupied mules. The objective is to define
the optimal deployment and task allocation strategy for the mules, so that the
sensors' downtime and the mules' traveling distance are minimized. Our
solutions are inspired by research in the field of computational geometry and
the design of our algorithms is based on state of the art approximation
algorithms for the classical problem of facility location. Our empirical
results demonstrate how cooperation enhances the team's performance, and
indicate that a combination of k-Median based deployment with closest-available
task allocation provides the best results in terms of minimizing the sensors'
downtime but is inefficient in terms of the mules' travel distance. A
k-Centroid based deployment produces good results in both criteria.Comment: 12 pages, 6 figures, conferenc
A Lightweight Distributed Solution to Content Replication in Mobile Networks
Performance and reliability of content access in mobile networks is
conditioned by the number and location of content replicas deployed at the
network nodes. Facility location theory has been the traditional, centralized
approach to study content replication: computing the number and placement of
replicas in a network can be cast as an uncapacitated facility location
problem. The endeavour of this work is to design a distributed, lightweight
solution to the above joint optimization problem, while taking into account the
network dynamics. In particular, we devise a mechanism that lets nodes share
the burden of storing and providing content, so as to achieve load balancing,
and decide whether to replicate or drop the information so as to adapt to a
dynamic content demand and time-varying topology. We evaluate our mechanism
through simulation, by exploring a wide range of settings and studying
realistic content access mechanisms that go beyond the traditional
assumptionmatching demand points to their closest content replica. Results show
that our mechanism, which uses local measurements only, is: (i) extremely
precise in approximating an optimal solution to content placement and
replication; (ii) robust against network mobility; (iii) flexible in
accommodating various content access patterns, including variation in time and
space of the content demand.Comment: 12 page
Tight Analysis of a Multiple-Swap Heuristic for Budgeted Red-Blue Median
Budgeted Red-Blue Median is a generalization of classic -Median in that
there are two sets of facilities, say and , that can
be used to serve clients located in some metric space. The goal is to open
facilities in and facilities in for
some given bounds and connect each client to their nearest open
facility in a way that minimizes the total connection cost.
We extend work by Hajiaghayi, Khandekar, and Kortsarz [2012] and show that a
multiple-swap local search heuristic can be used to obtain a
-approximation for Budgeted Red-Blue Median for any constant
. This is an improvement over their single swap analysis and
beats the previous best approximation guarantee of 8 by Swamy [2014].
We also present a matching lower bound showing that for every ,
there are instances of Budgeted Red-Blue Median with local optimum solutions
for the -swap heuristic whose cost is
times the optimum solution cost. Thus, our analysis is tight up to the lower
order terms. In particular, for any we show the single-swap
heuristic admits local optima whose cost can be as bad as times
the optimum solution cost
Approximation algorithms for stochastic clustering
We consider stochastic settings for clustering, and develop provably-good
approximation algorithms for a number of these notions. These algorithms yield
better approximation ratios compared to the usual deterministic clustering
setting. Additionally, they offer a number of advantages including clustering
which is fairer and has better long-term behavior for each user. In particular,
they ensure that *every user* is guaranteed to get good service (on average).
We also complement some of these with impossibility results
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