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Uniform Capacitated Clustering with Few Extra Centers — arXiv:2610.40625

Source

arXiv (en)

Original

https://arxiv.org/abs/2610.40625

License (as archived)

CC0-1.0

Archived

2026-11-01T02:47:50.702Z

SHA-256 of the archived text

c6ef684a964e353463b9f1d0d7a50810430510cf7eedb1d67d1acec655ea56ed

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How to cite

Uniform Capacitated Clustering with Few Extra Centers — arXiv:2610.40625 — arXiv (en), https://arxiv.org/abs/2610.40625 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_76fdf6d313df (Metraipolis archive dc_76fdf6d313df, sha256 c6ef684a964e353463b9f1d0d7a50810430510cf7eedb1d67d1acec655ea56ed)

@misc{dc76fdf6d313df,
  title = {Uniform Capacitated Clustering with Few Extra Centers — arXiv:2610.40625},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.40625},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_76fdf6d313df; sha256 c6ef684a964e353463b9f1d0d7a50810430510cf7eedb1d67d1acec655ea56ed},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.40625. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In uniform capacitated $(k,z)$-clustering, we open $k$ centers and assign every client to a center so that no center serves more than $U$ clients, minimizing the sum of the $z$-th powers of the assignment distances; $k$-median and $k$-means are the cases $z=1$ and $z=2$. We study algorithms that may open $k+\lfloor\varepsilon k\rfloor$ centers for a fixed $\varepsilon\in(0,1]$ and must respect the capacities exactly. In low dimensions, we give deterministic algorithms that are, except for $k$-median in the plane, the first polynomial-time algorithms with $(1+\varepsilon)k$ centers whose ratio is bounded independently of $\varepsilon$. In Euclidean space of any dimension, with clients of bounded doubling dimension, the ratio is $\mathcal{B}_z-(\mathcal{B}_z-1)\varepsilon+\delta$ for every fixed $\delta>0$, where $\mathcal{B}_z<3^z$ for $z>1$; this is $3-2\varepsilon+\delta$ for $k$-median and $5-4\varepsilon+\delta$ for $k$-means. In doubling metrics, the ratio is $3^z-(3^z-1)\varepsilon+\delta$ when centers may share a location, and a constant otherwise, such as $\min\{7-4\varepsilon,\,3+2\ln(1/\varepsilon)\}+\delta$ for $k$-median. In general metrics, deleting centers greedily gives ratio $3+2\ln(1/\varepsilon)+\delta$ for $k$-median when centers may share a location and $7+2\ln(1/\varepsilon)+\delta$ otherwise, improving the previous $O(\varepsilon^{-2}\log(1/\varepsilon))$. Our main tool is a rescue lemma. Splitting a near-optimal uncapacitated solution gives at most $2k-1$ blocks of at most $U$ clients, and giving a block a private center saves its share of the extra cost of following an optimal solution. Since a center of the optimal solution can be closed once all its blocks are rescued, $\lfloor\varepsilon k\rfloor$ extra centers save almost an $\varepsilon$-fraction of this extra cost instead of an $\varepsilon/2$-fraction.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv cs.DS, nuovi lavori del giorno).
Categorie arXiv: Computational Geometry (cs.CG) ; Data Structures and Algorithms (cs.DS).
Pagina del lavoro: https://arxiv.org/abs/2610.40625 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-01T02:17:58.616Z dalla pagina https://arxiv.org/list/cs.DS/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).