Approximation Algorithms for Clustering and Facility Location Problems
Abstract
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 costeffective 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 wellu
nderstood, but we have muchless understanding of moresophisticated models that better model the realworld concerns. In this thesis, we focus on three models inspired by some realworld 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 facilitymovement and clientassignment costs. This abstracts facilitylocation 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 localsearch based approximation algorithm for this problem and
achieve the bestknown 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 8approximation algorithm due to [34] based on LProunding. Our results extend to the weighted generalization wherein each facility i has a
nonnegative weight w_i and the movement cost for i is w_i times the distance
traveled by i.
In Chapter 3, we consider a facilitylocation problem that we call the minimumload kfacility 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 minmax 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 nontrivial true approximation of any kind was known for this metric).
Complementing this, we prove that MLkFL is strongly NPhard on line metrics.
In Chapter 4, we consider clustering problems with nonuniform 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 lowerbounded minsumofradii with outliers (LBkSRO) problem, and minimizing the maximum radius, which yields the lowerbounded ksupplier with outliers (LBkSupO) problem. We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the nonoutlier version. These also constitute the first approximation bounds for the minsumofradii objective when we consider lower bounds and outliers separately. We obtain approximation factors of 5 and 3 respectively for LBkSupO and its nonoutlier version. These are the first approximation results for ksupplier with nonuniform lower bounds.
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Cite this work
Sara Ahmadian
(2017).
Approximation Algorithms for Clustering and Facility Location Problems. UWSpace.
http://hdl.handle.net/10012/11640
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