An optimal approximation algorithm for the rectilinearm-center problem

Ko, Ming-Tat; Lee, Richard C. T.; Chang, Jyun-Sheng

doi:10.1007/bf01840393

Cited by 13 publications

(13 citation statements)

References 5 publications

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“…The approximation factor was later improved in Khuller and Sussmann [1996]. See Gonzalez [1991] and Ko et al [1990] for other approximation algorithms.…”

Section: Algorithms For Geometric Optimizationmentioning

confidence: 97%

“…Recently, Agarwal and Procopiuc [1998] extended and simplified the technique by Hwang et al [1993b] to obtain an n O(p 1Ϫ1/d ) -time algorithm for computing a p-center of n points in ‫ޒ‬ d . Therefore, for a fixed value of p, the Euclidean p-center (and also the Euclid-5 Please see Feder and Greene [1988], Gonzalez [1985], Ko et al [1990], Maass [1986], Megiddo [1990], and Megiddo and Supowit [1984].…”

Section: Euclidean P-centermentioning

confidence: 99%

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Efficient algorithms for geometric optimization

1998

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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LP-type problems and their efficient solution. We then describe a wide range of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other query-type problems.

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“…The approximation factor was later improved in Khuller and Sussmann [1996]. See Gonzalez [1991] and Ko et al [1990] for other approximation algorithms.…”

Section: Algorithms For Geometric Optimizationmentioning

confidence: 97%

Section: Euclidean P-centermentioning

confidence: 99%

Efficient algorithms for geometric optimization

1998

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“…Thus, the question is whether the constant 3 is also best possible for our special instances of the p-suppliers model. Achieving a constant which is strictly smaller than 2 is certainly NP-hard for our models, since the constant 2 is best possible for the classical unweighted p-center problem on general graphs [23,24], and even for planar rectilinear instances [28]. 2-approximation algorithms for the classical weighted, discrete and continuous p-center problem on general graphs are given in [37,38].…”

Section: Questions Comments and Concluding Remarksmentioning

confidence: 99%

One-way and round-trip center location problems

Tamir

Halman

2005

Discrete Optimization

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In the classical p-center problem there is a set V of points (customers) in some metric space, and the objective is to locate p centers (servers), minimizing the maximum distance between a customer and his respective nearest server. In this paper we consider an extension, where each customer is associated with a set of existing depots or distribution stations he can use. The service of a customer consists of the travel of a server to some permissible depot, loading of some package (e.g., a spare part) at the depot, and the delivery of the package to the customer. This model is called the customer one-way problem. In the round-trip version of the problem, the service also includes the travel from the customer to the home base of the server. In both problems the customer cost of the service is a linear function of the distance travelled by the server. The objective is to locate p servers, minimizing the maximum customer cost (weighted distance travelled by the respective server).Since the classical p-center problem is NP-hard, so are the one-way and the round-trip models we study. We present efficient constant factor approximation algorithms for these problems on general networks. Turning to special networks, we prove that the one-way problem is strongly NP-hard even on path networks. We then present polynomial time algorithms for the round-trip problem on general tree networks. We also discuss the single center case, and provide polynomial time algorithms for general networks, tree networks and planar Euclidean and rectilinear metric spaces.

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“…The k-center problem has a greedy 2-approximation algorithm (Gonzalez 1985;Hochbaum & Shmoys 1986): select a first center arbitrarily, and iteratively select the other c − 1 points each time maximizing the distance to the previously selected centers. Approximation and hardness of approximation results for k-center under various distance metrics can be found in Agarwal & Procopiuc (1998);Feder & Greene (1988); Gonzalez (1985); Hochbaum & Shmoys (1986);Kariv & Hakimi (1979); Ko et al (1990). The k-median problem also has a constant factor approximation algorithm.…”

Section: K-center and K-median Clusteringmentioning

confidence: 99%

Private Approximation of Clustering and Vertex Cover

2009

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Private approximation of search problems deals with finding approximate solutions to search problems while disclosing as little information as possible. The focus of this work is on private approximation of the vertex cover problem and two well studied clustering problems -k-center and k-median. Vertex cover was considered in (Beimel, Carmi, Nissim, and Weinreb, STOC, 2006) and we improve their infeasibility results. Clustering algorithms are frequently applied to sensitive data, and hence are of interest in the contexts of secure computation and private approximation. We show that these problems do not admit private approximations, or even approximation algorithms that are allowed to leak a significant number of bits of information. For the vertex cover problem we show a tight infeasibility result: every algorithm that ρ(n)-approximates vertex-cover must leak Ω(n/ρ(n)) bits (where n is the number of vertices in the graph). For the clustering problems we prove that even approximation algorithms with a poor approximation ratio must leak Ω(n) bits (where n is the number of points in the instance). For these results we develop new proof techniques, which are simpler and more intuitive than those in Beimel et al., and yet allow for stronger infeasibility results. Our proofs rely on the hardness of the promise problem where a unique optimal solution exists (Valiant and Vazirani, Theoretical Computer Science, 1986), on the hardness of approximating witnesses for NP-hard problems (Kumar and Sivakumar, CCC, (1999) andFeige, Langberg, and Nissim, APPROX, (2000)), and on a simple random embedding of instances into bigger instances.

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An optimal approximation algorithm for the rectilinearm-center problem

Cited by 13 publications

References 5 publications

Efficient algorithms for geometric optimization

Efficient algorithms for geometric optimization

One-way and round-trip center location problems

Private Approximation of Clustering and Vertex Cover

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