Parameterized Approximation For Robust Clustering in Discrete Geometric Spaces

Authors Fateme Abbasi , Sandip Banerjee, Jarosław Byrka , Parinya Chalermsook , Ameet Gadekar, Kamyar Khodamoradi, Dániel Marx , Roohani Sharma, Joachim Spoerhase



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Author Details

Fateme Abbasi
  • University of Wrocław, Poland
Sandip Banerjee
  • IDSIA, USI-SUPSI, Lugano, Switzerland
Jarosław Byrka
  • University of Wrocław, Poland
Parinya Chalermsook
  • Aalto University, Finland
Ameet Gadekar
  • Bar-Ilan University, Ramat-Gan, Israel
Kamyar Khodamoradi
  • University of Regina, Canada
Dániel Marx
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Roohani Sharma
  • University of Bergen, Norway
Joachim Spoerhase
  • University of Sheffield, UK

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Fateme Abbasi, Sandip Banerjee, Jarosław Byrka, Parinya Chalermsook, Ameet Gadekar, Kamyar Khodamoradi, Dániel Marx, Roohani Sharma, and Joachim Spoerhase. Parameterized Approximation For Robust Clustering in Discrete Geometric Spaces. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.6

Abstract

We consider the well-studied Robust (k,z)-Clustering problem, which generalizes the classic k-Median, k-Means, and k-Center problems and arises in the domains of robust optimization [Anthony, Goyal, Gupta, Nagarajan, Math. Oper. Res. 2010] and in algorithmic fairness [Abbasi, Bhaskara, Venkatasubramanian, 2021 & Ghadiri, Samadi, Vempala, 2022]. Given a constant z ≥ 1, the input to Robust (k,z)-Clustering is a set P of n points in a metric space (M,δ), a weight function w: P → ℝ_{≥ 0} and a positive integer k. Further, each point belongs to one (or more) of the m many different groups S_1,S_2,…,S_m ⊆ P. Our goal is to find a set X of k centers such that max_{i ∈ [m]} ∑_{p ∈ S_i} w(p) δ(p,X)^z is minimized. 
Complementing recent work on this problem, we give a comprehensive understanding of the parameterized approximability of the problem in geometric spaces where the parameter is the number k of centers. We prove the following results: [(i)] 
1) For a universal constant η₀ > 0.0006, we devise a 3^z(1-η₀)-factor FPT approximation algorithm for Robust (k,z)-Clustering in discrete high-dimensional Euclidean spaces where the set of potential centers is finite. This shows that the lower bound of 3^z for general metrics [Goyal, Jaiswal, Inf. Proc. Letters, 2023] no longer holds when the metric has geometric structure. 
2) We show that Robust (k,z)-Clustering in discrete Euclidean spaces is (√{3/2}- o(1))-hard to approximate for FPT algorithms, even if we consider the special case k-Center in logarithmic dimensions. This rules out a (1+ε)-approximation algorithm running in time f(k,ε)poly(m,n) (also called efficient parameterized approximation scheme or EPAS), giving a striking contrast with the recent EPAS for the continuous setting where centers can be placed anywhere in the space [Abbasi et al., FOCS'23]. 
3) However, we obtain an EPAS for Robust (k,z)-Clustering in discrete Euclidean spaces when the dimension is sublogarithmic (for the discrete problem, earlier work [Abbasi et al., FOCS'23] provides an EPAS only in dimension o(log log n)). Our EPAS works also for metrics of sub-logarithmic doubling dimension.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Facility location and clustering
Keywords
  • Clustering
  • approximation algorithms
  • parameterized complexity

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References

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