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Universal algorithms for clustering problems

Publication ,  Conference
Ganesh, A; Maggs, BM; Panigrahi, D
Published in: Leibniz International Proceedings in Informatics, LIPIcs
July 1, 2021

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 ℓp-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.

Duke Scholars

Published In

Leibniz International Proceedings in Informatics, LIPIcs

DOI

ISSN

1868-8969

Publication Date

July 1, 2021

Volume

198

Related Subject Headings

  • 46 Information and computing sciences
 

Citation

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Ganesh, A., Maggs, B. M., & Panigrahi, D. (2021). Universal algorithms for clustering problems. In Leibniz International Proceedings in Informatics, LIPIcs (Vol. 198). https://doi.org/10.4230/LIPIcs.ICALP.2021.70
Ganesh, A., B. M. Maggs, and D. Panigrahi. “Universal algorithms for clustering problems.” In Leibniz International Proceedings in Informatics, LIPIcs, Vol. 198, 2021. https://doi.org/10.4230/LIPIcs.ICALP.2021.70.
Ganesh A, Maggs BM, Panigrahi D. Universal algorithms for clustering problems. In: Leibniz International Proceedings in Informatics, LIPIcs. 2021.
Ganesh, A., et al. “Universal algorithms for clustering problems.” Leibniz International Proceedings in Informatics, LIPIcs, vol. 198, 2021. Scopus, doi:10.4230/LIPIcs.ICALP.2021.70.
Ganesh A, Maggs BM, Panigrahi D. Universal algorithms for clustering problems. Leibniz International Proceedings in Informatics, LIPIcs. 2021.

Published In

Leibniz International Proceedings in Informatics, LIPIcs

DOI

ISSN

1868-8969

Publication Date

July 1, 2021

Volume

198

Related Subject Headings

  • 46 Information and computing sciences