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DOI: 10.4230/LIPIcs.MFCS.2020.39

Graph Clustering in All Parameter Regimes

Authors: Junhao Gan, David F. Gleich, Nate Veldt, Anthony Wirth, and Xin Zhang

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract

Resolution parameters in graph clustering control the size and structure of clusters formed by solving a parametric objective function. Typically there is more than one meaningful way to cluster a graph, and solving the same objective function for different resolution parameters produces clusterings at different levels of granularity, each of which can be meaningful depending on the application. In this paper, we address the task of efficiently solving a parameterized graph clustering objective for all values of a resolution parameter. Specifically, we consider a new analysis-friendly objective we call LambdaPrime, involving a parameter λ ∈ (0,1). LambdaPrime is an adaptation of LambdaCC, a significant family of instances of the Correlation Clustering (minimization) problem. Indeed, LambdaPrime and LambdaCC are closely related to other parameterized clustering problems, such as parametric generalizations of modularity. They capture a number of specific clustering problems as special cases, including sparsest cut and cluster deletion. While previous work provides approximation results for a single value of the resolution parameter, we seek a set of approximately optimal clusterings for all values of λ in polynomial time. More specifically, we show that when a graph has m edges and n nodes, there exists a set of at most m clusterings such that, for every λ ∈ (0,1), the family contains an optimal solution to the LambdaPrime objective. This bound is tight on star graphs. We obtain a family of O(log n) clusterings by solving the parametric linear programming (LP) relaxation of LambdaPrime at O(log n) λ values, and rounding each LP solution using existing approximation algorithms. We prove that this is asymptotically tight: for a certain class of ring graphs, for all values of λ, Ω(log n) feasible solutions are required to provide a constant-factor approximation for the LambdaPrime LP relaxation. To minimize the size of the clustering family, we further propose an algorithm that yields a family of solutions of a size no more than twice of the minimum LP-approximating family.

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Junhao Gan, David F. Gleich, Nate Veldt, Anthony Wirth, and Xin Zhang. Graph Clustering in All Parameter Regimes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gan_et_al:LIPIcs.MFCS.2020.39,
  author =	{Gan, Junhao and Gleich, David F. and Veldt, Nate and Wirth, Anthony and Zhang, Xin},
  title =	{{Graph Clustering in All Parameter Regimes}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{39:1--39:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.39},
  URN =		{urn:nbn:de:0030-drops-127065},
  doi =		{10.4230/LIPIcs.MFCS.2020.39},
  annote =	{Keywords: Graph Clustering, Algorithms, Parametric Linear Programming}
}

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DOI: 10.4230/LIPIcs.ISAAC.2018.44

Correlation Clustering Generalized

Authors: David F. Gleich, Nate Veldt, and Anthony Wirth

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract

We present new results for LambdaCC and MotifCC, two recently introduced variants of the well-studied correlation clustering problem. Both variants are motivated by applications to network analysis and community detection, and have non-trivial approximation algorithms. We first show that the standard linear programming relaxation of LambdaCC has a Theta(log n) integrality gap for a certain choice of the parameter lambda. This sheds light on previous challenges encountered in obtaining parameter-independent approximation results for LambdaCC. We generalize a previous constant-factor algorithm to provide the best results, from the LP-rounding approach, for an extended range of lambda. MotifCC generalizes correlation clustering to the hypergraph setting. In the case of hyperedges of degree 3 with weights satisfying probability constraints, we improve the best approximation factor from 9 to 8. We show that in general our algorithm gives a 4(k-1) approximation when hyperedges have maximum degree k and probability weights. We additionally present approximation results for LambdaCC and MotifCC where we restrict to forming only two clusters.

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David F. Gleich, Nate Veldt, and Anthony Wirth. Correlation Clustering Generalized. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 44:1-44:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gleich_et_al:LIPIcs.ISAAC.2018.44,
  author =	{Gleich, David F. and Veldt, Nate and Wirth, Anthony},
  title =	{{Correlation Clustering Generalized}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{44:1--44:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.44},
  URN =		{urn:nbn:de:0030-drops-99925},
  doi =		{10.4230/LIPIcs.ISAAC.2018.44},
  annote =	{Keywords: Correlation Clustering, Approximation Algorithms}
}

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Graph Clustering in All Parameter Regimes

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Correlation Clustering Generalized

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