Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems

Berman, Piotr; Karpiński, Marek; Lingas, Andrzej

doi:10.1007/s00453-011-9591-5

Cited by 2 publications

(4 citation statements)

References 15 publications

(37 reference statements)

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“…The algorithm starts off by initializing σ to σ. While there is a violation of Inequality (7), we perform the following steps.…”

Section: The Preprocessing Stepmentioning

confidence: 99%

“…Applications of Set Cover include placement of wireless sensors or antennas to serve clients, VLSI design, and image processing [7,19]. It is natural to consider capacity constraints that appear in many applications, for instance, an upper bound on the number of clients that can be served by an antenna.…”

Section: Introductionmentioning

confidence: 99%

“…The generalization where each element belongs to at most a bounded number f of sets has been studied in a sequence of works, culminating in [20,30]. Berman et al [7] have considered the "soft" capacitated version of the CSC problem that allows making multiple copies of input sets. Another closely related problem to the CSC problem is the so-called Replica Placement problem.…”

Section: Introductionmentioning

confidence: 99%

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Capacitated Covering Problems in Geometric Spaces

Bandyapadhyay

Bhowmick

Inamdar

et al. 2019

Discrete Comput Geom

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In this article, we consider the following capacitated covering problem. We are given a set P of n points and a set B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in B. We would like to compute a subset B ⊆ B of balls and assign each point in P to some ball in B that contains it, such that the number of points assigned to any ball is at most U . The objective function that we would like to minimize is the cardinality of B .We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for these problems. In fact, in the Euclidean setting, only (1 + ) factor expansion is sufficient for any > 0, with the approximation factor being a polynomial in 1/ . We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems. complexity theoretic assumptions [14]. The same is true for the MUC, as demonstrated by the following reduction from Set Cover. We take a ball of radius 1 corresponding to each set, and a point corresponding to each element. If an element is in a set, then the distance between the center of the corresponding ball and the point is 1. We consider the metric space induced by the centers and the points. It is easy to see that any solution for this instance of MUC directly gives a solution for the input instance of the general Set Cover, implying that for MUC, it is not possible to get any approximation guarantee better than the O(log n) bound for Set Cover.The MUC in fixed dimensional Euclidean spaces has been extensively studied. One interesting variant is when the allowed set B of balls consists of all unit balls. Hochbaum and Maass [19] gave a polynomial time approximation scheme (PTAS) for this using a grid shifting strategy. When B is an arbitrary finite set of balls, the problem seems to be much harder. An O(1) approximation algorithm in the 2-dimensional Euclidean plane was given by Brönnimann and Goodrich [9]. More recently, a PTAS was obatined by Mustafa and Ray [26]. In dimensions 3 and higher, the best known approximation guarantee is still O(log n). Motivated by this, Har-Peled and Lee [18] gave a PTAS for a bi-criteria version where the algorithm is allowed to expand the input balls by a (1 + ) fac...

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“…The algorithm starts off by initializing σ to σ. While there is a violation of Inequality (7), we perform the following steps.…”

Section: The Preprocessing Stepmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Capacitated Covering Problems in Geometric Spaces

Bandyapadhyay

Bhowmick

Inamdar

et al. 2019

Discrete Comput Geom

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“…Similarly, covering problems have applications in various fields including wireless and sensor networks, VLSI, and image processing [2,24,32,34,65,72,74].…”

Section: Introductionmentioning

confidence: 99%

Digging deeper into clustering and covering problems

Bandyapadhyay¹

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Clustering problems often arise in the fields like data mining, machine learning and computational biology to group a collection of objects into similar groups with respect to a similarity measure. For example, clustering can be used to group genes with related expression patterns. Covering problems are another important class of problems, where the task is to select a subset of objects from a larger set, such that the objects in the subset "cover" (or contain) a given set of elements. Covering problems have found applications in various fields including wireless and sensor networks, VLSI, and image processing. For example, covering can be used to find placement locations of the minimum number of mobile towers to serve all the customers of a region. In this dissertation, we consider an interesting collection of geometric clustering and covering problems, which are modeled as optimization problems. These problems are known to be NP-hard, i.e. no efficient algorithms are expected to be found for these problems that return optimal solutions. Thus, we focus our effort in designing efficient approximation algorithms for these problems that yield near-optimal solutions. In this work, we study three clustering problems: k-means, k-clustering and Non-Uniformk-center and one covering problem: Metric Capacitated Covering. k-means is one of the most studied clustering problems and probably the most frequently used clustering problem in practical applications. In this problem, we are given a set of points in an Euclidean space and we want to choose k center points from the same Euclidean space. Each input point is assigned to its nearest chosen center, v and points assigned to a center form a cluster. The cost per input point is the square of its distance from its nearest center. The total cost is the sum of the costs of the points. The goal is to choose k center points so that the total cost is minimized. We give a local search based algorithm for this problem that always returns a solution of cost within (1 + ε)-factor of the optimal cost for any ε > 0. However, our algorithm uses (1+ε)k center points. The best known approximation before our work was about 9 that uses exactly k centers. The result appears in Chapter 2. k-clustering is another popular clustering problem studied mainly by the theory community. In this problem, each cluster is represented by a ball in the input metric space. We would like to choose k balls whose union contains all the input points. The cost of each ball is its radius to the power α for some given paramater α ≥ 1. The total cost is the sum of the costs of the chosen k balls. The goal is to find k balls such that the total cost is minimized. We give a probabilistic metric partitioning based algorithm for this problem that always returns a solution of cost within (1 + ε)-factor of the optimal cost for any ε > 0. However, our algorithm uses (1 + ε)k balls, and the running time is quasi-polynomial. The best known approximation in polynomial time is c α that uses exactly k balls, where c is a const...

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Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems

Cited by 2 publications

References 15 publications

Capacitated Covering Problems in Geometric Spaces

Capacitated Covering Problems in Geometric Spaces

Digging deeper into clustering and covering problems

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