Abstract not found

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 we may build a facility (such as a warehouse), where the cost of building at location i is f i ; furthermore, there is a set of client locations (such as 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 c ij is incurred. The objective is to 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 ineq...

We present improved combinatorial approximation algorithms for the uncapacitated facility location and k-median problems. Two central ideas in most of our results are cost scaling and greedy improvement. We present a simple greedy local search algorithm which achieves an approximation ratio of 2:414 + in ~ O(n 2 =) time. This also yields a bicriteria approximation tradeoff of (1 +; 1+ 2=) for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible. Combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to Jain and Vazirani, we get an approximation ratio of 1.853 in ~ O(n 3 ) time. This is already very close to the approximation guarantee of the best known algorithm which is LP-based. Further, combined with the best known LP-based algorithm for facility location, we get a very slight improvement in the approximation factor for facility location, achieving 1.728....

We present the first constant-factor approximation algorithm for the metric k-median problem. The k-median problem is one of the most well-studied 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 k-median 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 3-approximation 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

The dominating set problem asks for a small subset D of nodes in a graph such that every node is either in D or adjacent to a node in D. This problem arises in a number of distributed network applications, where it is important to locate a small number of centers in the network such that every node is nearby at least one center. Finding a dominating set of minimum size is NP-complete, and the best known approximation is logarithmic in the maximum degree of the graph and is provided by the same simple greedy approach that gives the well-known logarithmic approximation result for the closely related set cover problem.

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 LPtype problems and their efficient solution. We then describe a variety 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.

Publishing data for analysis from a table containing personal records, while maintaining individual privacy, is a problem of increasing importance today. The traditional approach of de-identifying records is to remove identifying fields such as social security number, name etc. However, recent research has shown that a large fraction of the US population can be identified using non-key attributes (called quasi-identifiers) such as date of birth, gender, and zip code [15]. Sweeney [16] proposed the k-anonymity model for privacy where non-key attributes that leak information are suppressed or generalized so that, for every record in the modified table, there are at least k−1 other records having exactly the same values for quasi-identifiers. We propose a new method for anonymizing data records, where quasi-identifiers of data records are first clustered and then cluster centers are published. To ensure privacy of the data records, we impose the constraint

In this paper we present a n O(k 1\Gamma1=d ) time algorithm for solving the k-center problem in R d , under L1 and L 2 metrics. The algorithm extends to other metrics, and can be used to solve the discrete k-center problem, as well. We also describe a simple (1 + ffl)-approximation algorithm for the k-center problem, with running time O(n log k) + (k=ffl) O(k 1\Gamma1=d ) . Finally, we present a n O(k 1\Gamma1=d ) time algorithm for solving the L-capacitated k-center problem, provided that L = \Omega\Gamma n=k 1\Gamma1=d ) or L = O(1). We conclude with a simple approximation algorithm for the L-capacitated k-center problem. The work on this paper was partially supported by a National Science Foundation Grant CCR-93--01259, by an Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award and matching funds from Xerox Corporation, and by a grant from the U.S.-Israeli Binational Science Foundation. y Department of Computer Science, Box ...

The capacitated K-center problem is a fundamental facility location problem, where we are asked to locate K facilities in a graph, and to assign vertices to facilities, so as to minimize the maximum distance from a vertex to the facility to which it is assigned. Moreover, each facility may be assigned at most L vertices. This problem is known to be NP-hard. We give polynomial time approximation algorithms for two different versions of this problem that achieve approximation factors of 5 and 6. We also study some generalizations of this problem. 1. Introduction The basic K-center problem is a fundamental facility location problem [17] and is defined as follows: given an edge-weighted graph G = (V; E) find a subset S ` V of size at most K such that each vertex in V is "close" to some vertex in S. More formally, the objective function is defined as follows: min S`V max u2V min v2S d(u; v) where d is the distance function. For example, one may wish to install K fire stations and mi...

Abstract not found