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10
A constantfactor approximation algorithm for the kmedian problem
 In Proceedings of the 31st Annual ACM Symposium on Theory of Computing
, 1999
"... We present the first constantfactor approximation algorithm for the metric kmedian problem. The kmedian problem is one of the most wellstudied clustering problems, i.e., those problems in which the aim is to partition a given set of points into clusters so that the points within a cluster are re ..."
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Cited by 215 (14 self)
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We present the first constantfactor approximation algorithm for the metric kmedian problem. The kmedian problem is one of the most wellstudied clustering problems, i.e., those problems in which the aim is to partition a given set of points into clusters so that the points within a cluster are relatively close with respect to some measure. For the metric kmedian problem, we are given n points in a metric space. We select k of these to be cluster centers, and then assign each point to its closest selected center. If point j is assigned to a center i, the cost incurred is proportional to the distance between i and j. The goal is to select the k centers that minimize the sum of the assignment costs. We give a 6 2 3approximation algorithm for this problem. This improves upon the best previously known result of O(log k log log k), which was obtained by refining and derandomizing a randomized O(log n log log n)approximation algorithm of Bartal. 1
A new greedy approach for facility location problems
"... We present a simple and natural greedy algorithm for the metric uncapacitated facility location problem achieving an approximation guarantee of 1.61 whereas the best previously known was 1.73. Furthermore, we will show that our algorithm has a property which allows us to apply the technique of Lagra ..."
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Cited by 116 (9 self)
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We present a simple and natural greedy algorithm for the metric uncapacitated facility location problem achieving an approximation guarantee of 1.61 whereas the best previously known was 1.73. Furthermore, we will show that our algorithm has a property which allows us to apply the technique of Lagrangian relaxation. Using this property, we can nd better approximation algorithms for many variants of the facility location problem, such as the capacitated facility location problem with soft capacities and a common generalization of the kmedian and facility location problem. We will also prove a lower bound on the approximability of the kmedian problem.
Greedy Facility Location Algorithms analyzed using Dual Fitting with FactorRevealing LP
 Journal of the ACM
, 2001
"... We present a natural greedy algorithm for the metric uncapacitated facility location problem and use the method of dual fitting to analyze its approximation ratio, which turns out to be 1.861. The running time of our algorithm is O(m log m), where m is the total number of edges in the underlying c ..."
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Cited by 100 (13 self)
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We present a natural greedy algorithm for the metric uncapacitated facility location problem and use the method of dual fitting to analyze its approximation ratio, which turns out to be 1.861. The running time of our algorithm is O(m log m), where m is the total number of edges in the underlying complete bipartite graph between cities and facilities. We use our algorithm to improve recent results for some variants of the problem, such as the fault tolerant and outlier versions. In addition, we introduce a new variant which can be seen as a special case of the concave cost version of this problem.
On the competitive ratio for online facility location
 In ICALP
, 2003
"... Abstract. We consider the problem of Online Facility Location, where demands arrive online and must be irrevocably assigned to an open facility upon arrival. The objective is to minimize the sum of facility and assignment costs. We prove that the competitive log n ratio for Online Facility Location ..."
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Cited by 15 (1 self)
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Abstract. We consider the problem of Online Facility Location, where demands arrive online and must be irrevocably assigned to an open facility upon arrival. The objective is to minimize the sum of facility and assignment costs. We prove that the competitive log n ratio for Online Facility Location is Θ (). On the negative side, we show that no log log n randomized algorithm can achieve a competitive ratio better than Ω ( ) against an log log n oblivious adversary even if the demands lie on a line segment. On the positive side, we present log n a deterministic algorithm achieving a competitive ratio of O (). The analysis is based log log n on a hierarchical decomposition of the optimal facility locations such that each component either is relatively wellseparated or has a relatively large diameter, and a potential function argument which distinguishes between the two kinds of components. 1
Approximating kmedian with nonuniform capacities
 In SODA ’05
, 2005
"... In this paper we give a constant factor approximation algorithm for the capacitated kmedian problem. Our algorithm produces a solution where capacities are exceeded by at most a constant factor, while the number of open facilities is at most k. This problem resisted attempts to apply the plethora o ..."
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Cited by 10 (0 self)
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In this paper we give a constant factor approximation algorithm for the capacitated kmedian problem. Our algorithm produces a solution where capacities are exceeded by at most a constant factor, while the number of open facilities is at most k. This problem resisted attempts to apply the plethora of methods designed for the uncapacitated case. Our algorithm is based on adding some new ingredients to the approach using the primaldual schema and lagrangian relaxations. Previous results on the capacitated kmedian problem gave approximations where the number of facilities is exceeded by some constant factor. Relaxing the constraint on the number of facilities seems to render kmedian problems much simpler. In some applications it is important not to violate the constraint on the number of facilities, whereas relaxing the capacity constraints is a natural thing to do, as the capacities express rough estimates on cluster sizes. 1
Improved Combinatorial Approximation Algorithms for the kLevel Facility Location Problem
, 2003
"... In this paper we present improved combinatorial approximation algorithms for the klevel facility location problem. First, by modifying the path reduction developed in [2], we obtain a combinatorial algorithm with a performance factor of 3:27 for any k 2, thus improving the previous bound of 4: ..."
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Cited by 4 (1 self)
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In this paper we present improved combinatorial approximation algorithms for the klevel facility location problem. First, by modifying the path reduction developed in [2], we obtain a combinatorial algorithm with a performance factor of 3:27 for any k 2, thus improving the previous bound of 4:56 achieved by a combinatorial algorithm. Then we develop another combinatorial algorithm that has a better performance guarantee and uses the rst algorithm as a subroutine. The latter algorithm can be recursively implemented and achieves a guarantee factor h(k), where h(k) is strictly less than 3:27 for any k and tends to 3:27 as k goes to 1. The values of h(k) can be easily computed with an arbitrary accuracy: h(2) 2:4211, h(3) 2:8446, h(4) 3:0565, h(5) 3:1678 and so on. Thus, for the cases of k = 2 and k = 3 the second combinatorial algorithm ensures an approximation factor substantially better than 3, which is currently the best approximation ratio for the klevel problem provided by the noncombinatorial algorithm due to Aardal, Chudak, and Shmoys [1].
Incremental algorithms for facility location and kmedian
 In Proceedings of the 12th Annual European Symposium on Algorithms, volume 3221 of Lecture Notes Comput. Sci
, 2004
"... In the incremental versions of Facility Location and kMedian, the demand points arrive one at a time and the algorithm maintains a good solution by either adding each new demand to an existing cluster or placing it in a new singleton cluster. The algorithm can also merge some of the existing cluste ..."
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Cited by 4 (0 self)
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In the incremental versions of Facility Location and kMedian, the demand points arrive one at a time and the algorithm maintains a good solution by either adding each new demand to an existing cluster or placing it in a new singleton cluster. The algorithm can also merge some of the existing clusters at any point in time. For Facility Location, we consider the case of uniform facility costs, where the cost of opening a facility is the same for all points, and present the first incremental algorithm which achieves a constant performance ratio. Using this algorithm as a building block, we obtain the first incremental algorithm for kMedian which achieves a constant performance ratio using O(k) medians. The algorithm is based on a novel merge rule which ensures that the algorithm’s configuration monotonically converges to the optimal facility locations according to a certain notion of distance. Using this property, we reduce the general case to the special case when the optimal solution consists of a single facility.
Dr. H. van Maaren
"... The development of MiniZSat did not happen overnight. Before starting with the actual development, research had to be done on both the Boolean satisfiability problem (SAT) and integer linear programming. Especially the complete understanding of SAT techniques took some time. Later on the pace picked ..."
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The development of MiniZSat did not happen overnight. Before starting with the actual development, research had to be done on both the Boolean satisfiability problem (SAT) and integer linear programming. Especially the complete understanding of SAT techniques took some time. Later on the pace picked up and MiniZSat was born. We initially started with the facility location problem and the idea to improve performance on this specific problem by using a SAT solver as a basis. During the process we realized that this was very difficult and we tried to create a general purpose semi pseudoBoolean solver. This resulted in the solver MiniZSat. The facility location problem still remains an integral part of the project. It became a case study of this thesis and it is extensively used to compare performance of various solvers. MiniZSat is a product of Hans van Maaren, Marijn Heule and myself. I want to thank them for their great assistance and suggestions while working on my project. They guided and supported me to create the MiniZSat solver and deliver this thesis. I also would like to thank my family and friends for their support through out this period. This thesis explores the possibility to increase the performance of a SATbased pseudoBoolean solver.