The Non-Uniform
            <i>k</i>
            -Center Problem

Chakrabarty, Deeparnab; Goyal, Prachi; Krishnaswamy, Ravishankar

doi:10.1145/3392720

Cited by 46 publications

(87 citation statements)

References 20 publications

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“…of Firefighters problem, which shows that NUkC is hard to approximate within a factor of γ in tree metric for any γ. We note that this is a strengthening of the result due to Chakrabarty et al [33] that proves only a constant inapproximabilty of the problem. Our reduction is however similar to the reduction in [33].…”

Section: Our Results and Contributionssupporting

confidence: 83%

“…We note that this is a strengthening of the result due to Chakrabarty et al [33] that proves only a constant inapproximabilty of the problem. Our reduction is however similar to the reduction in [33]. Then, we argue that the constructed tree instances of NUkC are √ γ-perturbation-resilient, and hence the similar hardness follows even for NUkC under √ γ-perturbation-resilience.…”

Section: Our Results and Contributionssupporting

confidence: 83%

“…For the Euclidean version of NUkC, we prove that it is not possible to get any "good" approximation for the problem. To prove the result, first we strengthen the result of Chakrabarty et al [33] to show that NUkC is hard to approximate within a factor of γ in tree metric for any γ. Then, we reduce the NUkC problem in tree metric to Euclidean NUkC.…”

Section: Our Results and Contributionssupporting

confidence: 52%

“…Using a reduction from the Firefighters problem [5], Chakrabarty et al [33] showed that the problem is NP-hard to approximate within any constant factor even in tree metric. On the other hand, they have designed a (c 1 , c 2 ) "bi-criteria" approximation for the problem for large constants c 1 and c 2 , i.e., if the algorithm is allowed to use c 1 balls of each type (thus c 1 •k in total), it can produce a solution with dilation at most c 2 times the optimal dilation.…”

Section: Previous Workmentioning

confidence: 99%

“…, r k , and the goal is to find the minimum dilation α and to choose k balls centered at the points of P with radius α • r i for 1 ≤ i ≤ k, such that all points of P are contained in the union of the chosen balls. The problem was formulated by Chakrabarty et al [33] as a generalization of the well-studied k-center clustering problem where all r i 's are the same. Due to the discrepancy in the radii of the balls, NUkC might be able to find hidden patterns in the data, which would not be possible for k-center (see Figure 1.4).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Our Results and Contributionssupporting

confidence: 83%

Section: Our Results and Contributionssupporting

confidence: 83%

Section: Our Results and Contributionssupporting

confidence: 52%

Section: Previous Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Digging deeper into clustering and covering problems

Bandyapadhyay¹

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A Technique for Obtaining True Approximations for k-Center with Covering Constraints

Anegg

Angelidakis

Kurpisz

et al. 2020

Integer Programming and Combinatorial Optimization

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There has been a recent surge of interest in incorporating fairness aspects into classical clustering problems. Two recently introduced variants of the k-Center problem in this spirit are Colorful k-Center, introduced by Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the Fair Robust k-Center problem introduced by Harris, Pensyl, Srinivasan, and Trinh. To address fairness aspects, these models, compared to traditional k-Center, include additional covering constraints. Prior approximation results for these models require to relax some of the normally hard constraints, like the number of centers to be opened or the involved covering constraints, and therefore, only obtain constant-factor pseudo-approximations. In this paper, we introduce a new approach to deal with such covering constraints that leads to (true) approximations, including a 4-approximation for Colorful k-Center with constantly many colorssettling an open question raised by Bandyapadhyay, Inamdar, Pai, and Varadarajan-and a 4-approximation for Fair Robust k-Center, for which the existence of a (true) constant-factor approximation was also open. We complement our results by showing that if one allows an unbounded number of colors, then Colorful k-Center admits no approximation algorithm with finite approximation guarantee, assuming that P = NP. Moreover, under the Exponential Time Hypothesis, the problem is inapproximable if the number of colors grows faster than logarithmic in the size of the ground set.

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Robust k-Center with Two Types of Radii

Chakrabarty

Negahbani

2021

Integer Programming and Combinatorial Optimization

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The Non-Uniform k -Center Problem

Cited by 46 publications

References 20 publications

Digging deeper into clustering and covering problems

Digging deeper into clustering and covering problems

A Technique for Obtaining True Approximations for k-Center with Covering Constraints

Robust k-Center with Two Types of Radii

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