Robust k-Center with Two Types of Radii

Chakrabarty, Deeparnab; Negahbani, Maryam

doi:10.1007/978-3-030-73879-2_19

Cited by 3 publications

(13 citation statements)

References 16 publications

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“…Next, we use the round-or-cut framework methodology from [5] on the instance I ′ , as described in Section 6. Essentially, this is a Turing reduction from Robust 3-NUkC to (polynomially many instances of) Well-Separated Robust 3-NUkC.…”

Section: Main Algorithm For 4-nukcmentioning

confidence: 99%

“…Chakrabarty et al [6] gave a bicriteria approximation for t-NUkC for arbitrary t, i.e., they give a solution containing O(k i ) balls of radius O(r i ) for 1 ≤ i ≤ t. They also give a (1 + √ 5)-approximation for 2-NUkC. Furthermore, they conjectured that there exists a polynomial time O(1)-approximation for t-NUkC for constant t. Subsequently, Chakrabarty and Negahbani [5] made some progress by giving a 10-approximation for Robust 2-NUkC. Very recently, Jia et al [13] showed an approximate equivalence between (t + 1)-NUkC and Robust t-NUkC, thereby observing that the previous result of [5] readily implies a 23-approximation for 3-NUkC.…”

Section: Introductionmentioning

confidence: 98%

“…Furthermore, they conjectured that there exists a polynomial time O(1)-approximation for t-NUkC for constant t. Subsequently, Chakrabarty and Negahbani [5] made some progress by giving a 10-approximation for Robust 2-NUkC. Very recently, Jia et al [13] showed an approximate equivalence between (t + 1)-NUkC and Robust t-NUkC, thereby observing that the previous result of [5] readily implies a 23-approximation for 3-NUkC. We note that the techniques from Inamdar and Varadarajan [11] implicitly give an O(1)-approximation for t-NUkC for any t ≥ 1, in k O(k) • n O (1) time, where k = t k t .…”

Section: Introductionmentioning

confidence: 99%

“…First, we reduce the 4-NUkC problem to the Robust 3-NUkC problem, following Jia et al [13]. Next, we reduce the Robust 3-NUkC to well-separated Robust 3-NUkC, by adapting the approach of Chakrabarty and Negahbani [5]. 1 In a well-separated instance, we are given a set of potential centers for the balls of radius r 1 , such that the distance between any two of these centers is at least c • r 1 , for a parameter c ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

“…Our overall algorithm for 4-NUkC is combinatorial, except for the step where we reduce Robust 3-NUkC to Well-Separated Robust 3-NUkC using the round-or-cut approach of [5]. Thus, we avoid an additional "inner loop" of round-or-cut that is employed in recent work [5,13]. 2 2 Definitions, Main Result, and Greedy Clustering…”

Section: Introductionmentioning

confidence: 99%

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Non-Uniform $k$-Center and Greedy Clustering

Inamdar¹,

Varadarajan²

2021

Preprint

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In the Non-Uniform k-Center (NUkC) problem, a generalization of the famous k-center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In t-NUkC, we assume that the number of distinct radii is equal to t, and we are allowed to use k i balls of radius r i , for 1 ≤ i ≤ t. This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for t-NUkC is not possible if t is unbounded. On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for t-NUkC should be possible if t is a fixed constant. Since then, there has been steady progress towards resolving this conjecture -currently, a constant approximation for 3-NUkC is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [To appear in SOSA 2022]. We push the horizon by giving an O(1)-approximation for the Non-Uniform k-Center for 4 distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the k-center literature, which also demonstrates that the different generalizations of kcenter involving non-uniform radii, and multiple coverage constraints (i.e., colorful k-center ), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the t-NUkC problem, eventually bringing us closer to the resolution of the CGK conjecture.

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Section: Main Algorithm For 4-nukcmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Non-Uniform $k$-Center and Greedy Clustering

Inamdar¹,

Varadarajan²

2021

Preprint

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The Nearest Neighbor Algorithm for Balanced and Connected k-Center Problem under Modular Distance

Li¹,

Yang²

2021

2021 International Conference on Networking and Network Applications (NaNA)

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Towards Non-Uniform k-Center with Constant Types of Radii

Jia¹,

Rohwedder²,

Sheth³

et al. 2021

Preprint

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In the Non-Uniform k-Center problem we need to cover a finite metric space using k balls of different radii that can be scaled uniformly. The goal is to minimize the scaling factor. If the number of different radii is unbounded, the problem does not admit a constant-factor approximation algorithm but it has been conjectured that such an algorithm exists if the number of radii is constant. Yet, this is known only for the case of two radii. Our first contribution is a simple black box reduction which shows that if one can handle the variant of t − 1 radii with outliers, then one can also handle t radii. Together with an algorithm by Chakrabarty and Negahbani for two radii with outliers, this immediately implies a constant-factor approximation algorithm for three radii; thus making further progress on the conjecture. Furthermore, using algorithms for the k-center with outliers problem, that is the one radii with outliers case, we also get a simple algorithm for two radii.The algorithm by Chakrabarty and Negahbani uses a top-down approach, starting with the larger radius and then proceeding to the smaller one. Our reduction, on the other hand, looks only at the smallest radius and eliminates it, which suggests that a bottom-up approach is promising. In this spirit, we devise a modification of the Chakrabarty and Negahbani algorithm which runs in a bottom-up fashion, and in this way we recover their result with the advantage of having a simpler analysis.

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Robust k-Center with Two Types of Radii

Cited by 3 publications

References 16 publications

Non-Uniform $k$-Center and Greedy Clustering

Non-Uniform $k$-Center and Greedy Clustering

The Nearest Neighbor Algorithm for Balanced and Connected k-Center Problem under Modular Distance

Towards Non-Uniform k-Center with Constant Types of Radii

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