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On the equivalence between the primaldual schema and the local ratio technique
 In 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX). Number 2129 in Lecture Notes in Computer Science
, 2001
"... Abstract. We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primaldual schema and the local ratio technique. Recently, primaldual algorithms were devised by first constructing a local ratio algorithm and then transform ..."
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Abstract. We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primaldual schema and the local ratio technique. Recently, primaldual algorithms were devised by first constructing a local ratio algorithm and then transforming it into a primaldual algorithm. This was done in the case of the 2approximation algorithms for the feedback vertex set problem and in the case of the first primaldual algorithms for maximization problems. Subsequently, the nature of the connection between the two paradigms was posed as an open question by Williamson [Math. Program., 91 (2002), pp. 447–478]. In this paper we answer this question by showing that the two paradigms are equivalent.
Geometric clustering to minimize the sum of cluster sizes
 In Proc. 13th European Symp. Algorithms, Vol 3669 of LNCS
, 2005
"... Abstract. We study geometric versions of the minsize kclustering problem, a clustering problem which generalizes clustering to minimize the sum of cluster radii and has important applications. We prove that the problem can be solved in polynomial time when the points to be clustered are located on ..."
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Abstract. We study geometric versions of the minsize kclustering problem, a clustering problem which generalizes clustering to minimize the sum of cluster radii and has important applications. We prove that the problem can be solved in polynomial time when the points to be clustered are located on a line. For Euclidean spaces of higher dimensions, we show that the problem is NPhard and present polynomial time approximation schemes. The latter result yields an improved approximation algorithm for the related problem of kclustering to minimize the sum of cluster diameters. 1
A note on multicovering with disks
 Comput. Geom
"... In theDisk Multicover problem the input consists of a set P of n points in the plane, where each point p ∈ P has a covering requirement k(p), and a set B of m base stations, where each base station b ∈ B has a weight w(b). If a base station b ∈ B is assigned a radius r(b), it covers all points in th ..."
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In theDisk Multicover problem the input consists of a set P of n points in the plane, where each point p ∈ P has a covering requirement k(p), and a set B of m base stations, where each base station b ∈ B has a weight w(b). If a base station b ∈ B is assigned a radius r(b), it covers all points in the disk of radius r(b) centered at b. The weight of a radii assignment r: B → R is defined as b∈B w(b)r(b) α, for some constant α. A feasible solution is an assignment such that each point p is covered by at least k(p) disks, and the goal is to find a minimum weight feasible solution. The Polygon Disk Multicover problem is a closely related problem, in which the set P is a polygon (possibly with holes), and the goal is to find a minimum weight radius assignment that covers each point in P at least K times. We present a 3αkmaxapproximation algorithm for Disk Multicover, where kmax is the maximum covering requirement of a point, and a (3αK + ε)approximation algorithm for Polygon Disk Multicover.
A constantfactor approximation for multicovering with disks
 In Proceedings of the Symposium on Computational Geometry (SoCG
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Journal of Computational Geometry jocg.org A CONSTANTFACTOR APPROXIMATION FOR MULTICOVERING WITH DISKS∗
"... Abstract. We consider the following multicovering problem with disks. We are given two point sets Y (servers) and X (clients) in the plane, a coverage function κ: X → N, and a constant α ≥ 1. Centered at each server is a single disk whose radius we are free to set. The requirement is that each clie ..."
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Abstract. We consider the following multicovering problem with disks. We are given two point sets Y (servers) and X (clients) in the plane, a coverage function κ: X → N, and a constant α ≥ 1. Centered at each server is a single disk whose radius we are free to set. The requirement is that each client x ∈ X be covered by at least κ(x) of the server disks. The objective function we wish to minimize is the sum of the αth powers of the disk radii. We present a polynomialtime algorithm for this problem achieving an O(1) approximation. 1
A PTAS for the disk cover problem of geometric objects
"... We present PTASes for the disk cover problem: given geometric objects and a finite set of disk centers, minimize the total cost for covering those objects with disks under a polynomial cost function on the disks ’ radii. We describe the first FPTAS for covering a line segment when the disk centers f ..."
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We present PTASes for the disk cover problem: given geometric objects and a finite set of disk centers, minimize the total cost for covering those objects with disks under a polynomial cost function on the disks ’ radii. We describe the first FPTAS for covering a line segment when the disk centers form a discrete set, and a PTAS for when a set of geometric objects, described by polynomial algebraic inequalities, must be covered. The latter result holds for any dimension.