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Parameterized Approximation Schemes for Clustering with General Norm Objectives

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conference contribution
posted on 2024-10-30, 16:56 authored by Fateme Abbasi, Sandip Banerjee, Jarosław Byrka, Parinya Chalermsook, Ameet Gadekar, Kamyar Khodamoradi, Dániel MarxDániel Marx, Roohani Sharma, Joachim Spoerhase
This paper considers the well-studied algorithmic regime of designing a (1+ϵ)-approximation algorithm for a k-clustering problem that runs in time f(k,ϵ)poly(n) (sometimes called an efficient parameterized approximation scheme or EPAS for short). Notable results of this kind include EPASes in the high-dimensional Euclidean setting for k-center [Badŏiu, Har-Peled, Indyk; STOC'02] as well as k-median, and k-means [Kumar, Sabharwal, Sen; J. ACM 2010]. However, existing EPASes handle only basic objectives (such as k-center, k-median, and k-means) and are tailored to the specific objective and metric space. Our main contribution is a clean and simple EPAS that settles more than ten clustering problems (across multiple well-studied objectives as well as metric spaces) and unifies well-known EPASes. Our algorithm gives EPASes for a large variety of clustering objectives (for example, k-means, k-center, k-median, priority k-center, ℓ-centrum, ordered k-median, socially fair k-median aka robust k-median, or more generally monotone norm k-clustering) and metric spaces (for example, continuous high-dimensional Euclidean spaces, metrics of bounded doubling dimension, bounded treewidth metrics, and planar metrics). Key to our approach is a new concept that we call bounded ϵ-scatter dimension--an intrinsic complexity measure of a metric space that is a relaxation of the standard notion of bounded doubling dimension. Our main technical result shows that two conditions are essentially sufficient for our algorithm to yield an EPAS on the input metric M for any clustering objective: (i) The objective is described by a monotone (not necessarily symmetric!) norm, and (ii) the ϵ-scatter dimension of M is upper bounded by a function of ϵ.

History

Primary Research Area

  • Algorithmic Foundations and Cryptography

Name of Conference

IEEE Symposium on Foundations of Computer Science (FOCS)

Volume

00

Page Range

1377-1399

Publisher

Institute of Electrical and Electronics Engineers (IEEE)

Open Access Type

  • Green

BibTeX

@conference{Abbasi:Banerjee:Byrka:Chalermsook:Gadekar:Khodamoradi:Marx:Sharma:Spoerhase:2023, title = "Parameterized Approximation Schemes for Clustering with General Norm Objectives", author = "Abbasi, Fateme" AND "Banerjee, Sandip" AND "Byrka, Jarosław" AND "Chalermsook, Parinya" AND "Gadekar, Ameet" AND "Khodamoradi, Kamyar" AND "Marx, Dániel" AND "Sharma, Roohani" AND "Spoerhase, Joachim", year = 2023, month = 11, pages = "1377--1399", publisher = "Institute of Electrical and Electronics Engineers (IEEE)", doi = "10.1109/focs57990.2023.00085" }