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Faster Approximation Schemes for (Constrained) k-Means with Outliers

Authors Zhen Zhang , Junyu Huang , Qilong Feng

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Author Details

Zhen Zhang
  • School of Advanced Interdisciplinary Studies, Hunan University of Technology and Business, Changsha, China
  • Xiangjiang Laboratory, Changsha, China
Junyu Huang
  • School of Computer Science and Engineering, Central South University, Changsha, China
Qilong Feng
  • School of Computer Science and Engineering, Central South University, Changsha, China

Funding

This work was supported by National Natural Science Foundation of China (62202161, 62172446), Natural Science Foundation of Hunan Province (2023JJ40240), and Scientific Research Fund of Hunan Provincial Education Department (23B0597).

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Zhen Zhang, Junyu Huang, and Qilong Feng. Faster Approximation Schemes for (Constrained) k-Means with Outliers. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 84:1-84:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.84

Abstract

Given a set of n points in ℝ^d and two positive integers k and m, the Euclidean k-means with outliers problem aims to remove at most m points, referred to as outliers, and minimize the k-means cost function for the remaining points. Developing algorithms for this problem remains an active area of research due to its prevalence in applications involving noisy data. In this paper, we give a (1+ε)-approximation algorithm that runs in n²d((k+m)ε^{-1})^O(kε^{-1}) time for the problem. When combined with a coreset construction method, the running time of the algorithm can be improved to be linear in n. For the case where k is a constant, this represents the first polynomial-time approximation scheme for the problem: Existing algorithms with the same approximation guarantee run in polynomial time only when both k and m are constants. Furthermore, our approach generalizes to variants of k-means with outliers incorporating additional constraints on instances, such as those related to capacities and fairness.

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
Keywords
  • Approximation algorithms
  • clustering

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