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

Anegg, Georg; Angelidakis, Haris; Kurpisz, Adam; Zenklusen, Rico

doi:10.1007/978-3-030-45771-6_5

Cited by 28 publications

(35 citation statements)

References 22 publications

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“…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 . That is, one gets an FPT approximation.…”

Section: Introductionmentioning

confidence: 65%

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: Introductionmentioning

confidence: 65%

Non-Uniform $k$-Center and Greedy Clustering

Inamdar¹,

Varadarajan²

2021

Preprint

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“…While we show that individually they do not improve the integrality gap, we believe that their combination can lead to a strong relaxation. Independent work Independently and concurrently to our work, authors in [2] obtained a 4-approximation algorithm for the colorful k-center problem with ω = O(1) and running time |P| O(ω) using different techniques than the ones described in this work. Furthermore they show that, assuming P = N P, if ω is allowed to be unbounded then the colorful k-center problem admits no algorithm guaranteeing a finite approximation.…”

Section: Introductionmentioning

confidence: 95%

“…Now consider some k th index in the sequence, k > k where k is the dimension of M P t (W ) (w). By property (2), it is of the form J ∪ {x}, where J is one of the first k − 1 indices, and x ∈ V . There are two cases:…”

Section: Definitionmentioning

confidence: 99%

“…Our 3-approximation algorithm for colorful k-center with ω color classes runs in time |P| O(ω 2 ) , where the quadratic term arises from guessing linearly many optimal centers to make up for linearly many extra centers in the pseudo-approximation. In [2], it was shown that a linear exponential dependence on ω is necessary assuming ETH holds. It would be interesting to obtain the same approximation factor but without the quadratic dependence on ω, or better yet, obtain a tight result.…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

See 1 more Smart Citation

Fair colorful k-center clustering

2021

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An instance of colorfulk-center consists of points in a metric space that are colored red or blue, along with an integer k and a coverage requirement for each color. The goal is to find the smallest radius $$\rho $$ ρ such that there exist balls of radius $$\rho $$ ρ around k of the points that meet the coverage requirements. The motivation behind this problem is twofold. First, from fairness considerations: each color/group should receive a similar service guarantee, and second, from the algorithmic challenges it poses: this problem combines the difficulties of clustering along with the subset-sum problem. In particular, we show that this combination results in strong integrality gap lower bounds for several natural linear programming relaxations. Our main result is an efficient approximation algorithm that overcomes these difficulties to achieve an approximation guarantee of 3, nearly matching the tight approximation guarantee of 2 for the classical k-center problem which this problem generalizes. algorithms either opened more than k centers or only worked in the special case when the input points are in the plane.

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“…For the colorful k-center problem a constant approximation in the Euclidean plane was recently introduced (Bandyapadhyay et al 2019). In Anegg et al (2020), the authors study a variant of this problem in which classes are allowed to overlap. Neither of the proposed algorithms is directly applicable to scRNA-seq data, due to low-dimensionality assumptions or the use of the ellipsoid method, respectively.…”

Section: Fair Samplingmentioning

confidence: 99%

Sphetcher: Spherical Thresholding Improves Sketching of Single-Cell Transcriptomic Heterogeneity

2020

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The massive size of single-cell RNA sequencing datasets often exceeds the capability of current computational analysis methods to solve routine tasks such as detection of cell types. Recently, geometric sketching was introduced as an alternative to uniform subsampling. It selects a subset of cells (the sketch) that evenly cover the transcriptomic space occupied by the original dataset, to accelerate downstream analyses and highlight rare cell types. Here, we propose algorithm Sphetcher that makes use of the thresholding technique to efficiently pick representative cells within spheres (as opposed to the typically used equal-sized boxes) that cover the entire transcriptomic space. We show that the spherical sketch computed by Sphetcher constitutes a more accurate representation of the original transcriptomic landscape. Our optimization scheme allows to include fairness aspects that can encode prior biological or experimental knowledge. We show how a fair sampling can inform the inference of the trajectory of human skeletal muscle myoblast differentiation.

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

Cited by 28 publications

References 22 publications

Non-Uniform $k$-Center and Greedy Clustering

Non-Uniform $k$-Center and Greedy Clustering

Fair colorful k-center clustering

Sphetcher: Spherical Thresholding Improves Sketching of Single-Cell Transcriptomic Heterogeneity

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