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Document

DOI: 10.4230/LIPIcs.ISAAC.2024.16

FPT Approximations for Fair k-Min-Sum-Radii

Authors: Lena Carta, Lukas Drexler, Annika Hennes, Clemens Rösner, and Melanie Schmidt

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract

We consider the k-min-sum-radii (k-MSR) clustering problem with fairness constraints. The k-min-sum-radii problem is a mixture of the classical k-center and k-median problems. We are given a set of points P in a metric space and a number k and aim to partition the points into k clusters, each of the clusters having one designated center. The objective to minimize is the sum of the radii of the k clusters (where in k-center we would only consider the maximum radius and in k-median we would consider the sum of the individual points' costs). Various notions of fair clustering have been introduced lately, and we follow the definitions due to Chierichetti et al. [Flavio Chierichetti et al., 2017] which demand that cluster compositions shall follow the proportions of the input point set with respect to some given sensitive attribute. For the easier case where the sensitive attribute only has two possible values and each is equally frequent in the input, the aim is to compute a clustering where all clusters have a 1:1 ratio with respect to this attribute. We call this the 1:1 case. There has been a surge of FPT-approximation algorithms for the k-MSR problem lately, solving the problem both in the unconstrained case and in several constrained problem variants. We add to this research area by designing an FPT (6+ε)-approximation that works for k-MSR under the mentioned general fairness notion. For the special 1:1 case, we improve our algorithm to achieve a (3+ε)-approximation.

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Lena Carta, Lukas Drexler, Annika Hennes, Clemens Rösner, and Melanie Schmidt. FPT Approximations for Fair k-Min-Sum-Radii. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{carta_et_al:LIPIcs.ISAAC.2024.16,
  author =	{Carta, Lena and Drexler, Lukas and Hennes, Annika and R\"{o}sner, Clemens and Schmidt, Melanie},
  title =	{{FPT Approximations for Fair k-Min-Sum-Radii}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{16:1--16:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.16},
  URN =		{urn:nbn:de:0030-drops-221438},
  doi =		{10.4230/LIPIcs.ISAAC.2024.16},
  annote =	{Keywords: Clustering, k-min-sum-radii, fairness}
}

Artifact

Software

DOI: 10.4230/artifacts.22470

algo-hhu/FLSpp

Authors: Lukas Drexler, Joshua Könen, Daniel R. Schmidt, Melanie Schmidt, and Giulia Baldini

Abstract

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Lukas Drexler, Joshua Könen, Daniel R. Schmidt, Melanie Schmidt, Giulia Baldini. algo-hhu/FLSpp (Software). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@misc{dagstuhl-artifact-22470,
   title = {{algo-hhu/FLSpp}}, 
   author = {Drexler, Lukas and K\"{o}nen, Joshua and Schmidt, Daniel R. and Schmidt, Melanie and Baldini, Giulia},
   note = {Software, swhId: \href{https://archive.softwareheritage.org/swh:1:dir:39136777e542456572a51515813b6cb377ed0940;origin=https://github.com/algo-hhu/FLSpp;visit=swh:1:snp:18f736f0d6ee924ac77fcfdbeed3e15015a6b867;anchor=swh:1:rev:72d58dc26e599c190373274ed6c22009d07a911b}{\texttt{swh:1:dir:39136777e542456572a51515813b6cb377ed0940}} (visited on 2024-11-28)},
   url = {https://github.com/algo-hhu/FLSpp},
   doi = {10.4230/artifacts.22470},
}

Document

DOI: 10.4230/LIPIcs.SEA.2024.7

Local Search k-means++ with Foresight

Authors: Theo Conrads, Lukas Drexler, Joshua Könen, Daniel R. Schmidt, and Melanie Schmidt

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract

Since its introduction in 1957, Lloyd’s algorithm for k-means clustering has been extensively studied and has undergone several improvements. While in its original form it does not guarantee any approximation factor at all, Arthur and Vassilvitskii (SODA 2007) proposed k-means++ which enhances Lloyd’s algorithm by a seeding method which guarantees a 𝒪(log k)-approximation in expectation. More recently, Lattanzi and Sohler (ICML 2019) proposed LS++ which further improves the solution quality of k-means++ by local search techniques to obtain a 𝒪(1)-approximation. On the practical side, the greedy variant of k-means++ is often used although its worst-case behaviour is provably worse than for the standard k-means++ variant. We investigate how to improve LS++ further in practice. We study two options for improving the practical performance: (a) Combining LS++ with greedy k-means++ instead of k-means++, and (b) Improving LS++ by better entangling it with Lloyd’s algorithm. Option (a) worsens the theoretical guarantees of k-means++ but improves the practical quality also in combination with LS++ as we confirm in our experiments. Option (b) is our new algorithm, Foresight LS++. We experimentally show that FLS++ improves upon the solution quality of LS++. It retains its asymptotic runtime and its worst-case approximation bounds.

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Theo Conrads, Lukas Drexler, Joshua Könen, Daniel R. Schmidt, and Melanie Schmidt. Local Search k-means++ with Foresight. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{conrads_et_al:LIPIcs.SEA.2024.7,
  author =	{Conrads, Theo and Drexler, Lukas and K\"{o}nen, Joshua and Schmidt, Daniel R. and Schmidt, Melanie},
  title =	{{Local Search k-means++ with Foresight}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{7:1--7:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.7},
  URN =		{urn:nbn:de:0030-drops-203727},
  doi =		{10.4230/LIPIcs.SEA.2024.7},
  annote =	{Keywords: k-means clustering, kmeans++, greedy, local search}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2023.50

Connected k-Center and k-Diameter Clustering

Authors: Lukas Drexler, Jan Eube, Kelin Luo, Heiko Röglin, Melanie Schmidt, and Julian Wargalla

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract

Motivated by an application from geodesy, we study the connected k-center problem and the connected k-diameter problem. These problems arise from the classical k-center and k-diameter problems by adding a side constraint. For the side constraint, we are given an undirected connectivity graph G on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in G. Usually in clustering problems one assumes that the clusters are pairwise disjoint. We study this case but additionally also the case that clusters are allowed to be non-disjoint. This can help to satisfy the connectivity constraints. Our main result is an O(1)-approximation algorithm for the disjoint connected k-center and k-diameter problem for Euclidean spaces of low dimension (constant d) and for metrics with constant doubling dimension. For general metrics, we get an O(log²k)-approximation. Our algorithms work by computing a non-disjoint connected clustering first and transforming it into a disjoint connected clustering. We complement these upper bounds by several upper and lower bounds for variations and special cases of the model.

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Lukas Drexler, Jan Eube, Kelin Luo, Heiko Röglin, Melanie Schmidt, and Julian Wargalla. Connected k-Center and k-Diameter Clustering. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{drexler_et_al:LIPIcs.ICALP.2023.50,
  author =	{Drexler, Lukas and Eube, Jan and Luo, Kelin and R\"{o}glin, Heiko and Schmidt, Melanie and Wargalla, Julian},
  title =	{{Connected k-Center and k-Diameter Clustering}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{50:1--50:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.50},
  URN =		{urn:nbn:de:0030-drops-181024},
  doi =		{10.4230/LIPIcs.ICALP.2023.50},
  annote =	{Keywords: Approximation algorithms, Clustering, Connectivity constraints}
}

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Documents authored by Drexler, Lukas

FPT Approximations for Fair k-Min-Sum-Radii

Abstract

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algo-hhu/FLSpp

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Local Search k-means++ with Foresight

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Connected k-Center and k-Diameter Clustering

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