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Universal Algorithms for Clustering Problems

Authors Arun Ganesh, Bruce M. Maggs, Debmalya Panigrahi

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Author Details

Arun Ganesh
  • Department of Computer Science, University of California at Berkeley, CA, USA
Bruce M. Maggs
  • Department of Computer Science, Duke University, Durham, NC, USA
  • Emerald Innovations, Cambridge, MA, USA
Debmalya Panigrahi
  • Department of Computer Science, Duke University, Durham, NC, USA

Funding

  • Ganesh, Arun: Supported in part by NSF Award CCF-1535989.
  • Maggs, Bruce M.: Supported in part by NSF grant CCF-1535972.
  • Panigrahi, Debmalya: Supported in part by NSF grants CCF-1535972, CCF-1955703, an NSF Career Award CCF-1750140, and the Indo-US Virtual Networked Joint Center on Algorithms Under Uncertainty.

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Arun Ganesh, Bruce M. Maggs, and Debmalya Panigrahi. Universal Algorithms for Clustering Problems. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 70:1-70:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.70

Abstract

This paper presents universal algorithms for clustering problems, including the widely studied k-median, k-means, and k-center objectives. The input is a metric space containing all potential client locations. The algorithm must select k cluster centers such that they are a good solution for any subset of clients that actually realize. Specifically, we aim for low regret, defined as the maximum over all subsets of the difference between the cost of the algorithm’s solution and that of an optimal solution. A universal algorithm’s solution sol for a clustering problem is said to be an (α, β)-approximation if for all subsets of clients C', it satisfies sol(C') ≤ α ⋅ opt(C') + β ⋅ mr, where opt(C') is the cost of the optimal solution for clients C' and mr is the minimum regret achievable by any solution. Our main results are universal algorithms for the standard clustering objectives of k-median, k-means, and k-center that achieve (O(1), O(1))-approximations. These results are obtained via a novel framework for universal algorithms using linear programming (LP) relaxations. These results generalize to other 𝓁_p-objectives and the setting where some subset of the clients are fixed. We also give hardness results showing that (α, β)-approximation is NP-hard if α or β is at most a certain constant, even for the widely studied special case of Euclidean metric spaces. This shows that in some sense, (O(1), O(1))-approximation is the strongest type of guarantee obtainable for universal clustering.

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
Keywords
  • universal algorithms
  • clustering
  • k-median
  • k-means
  • k-center

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