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On 2-Clubs in Graph-Based Data Clustering: Theory and Algorithm Engineering

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Algorithms and Complexity (CIAC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12701))

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Abstract

Editing a graph into a disjoint union of clusters is a standard optimization task in graph-based data clustering. Here, complementing classic work where the clusters shall be cliques, we focus on clusters that shall be 2-clubs, that is, subgraphs of diameter at most two. This naturally leads to the two NP-hard problems 2-Club Cluster Editing (the editing operations are edge insertion and edge deletion) and 2-Club Cluster Vertex Deletion (the editing operations are vertex deletions). Answering an open question, we show that 2-Club Cluster Editing is W[2]-hard with respect to the number of edge modifications, thus contrasting the fixed-parameter tractability result for the classic Cluster Editing problem (considering cliques instead of 2-clubs). Then, focusing on 2-Club Cluster Vertex Deletion, which is easily seen to be fixed-parameter tractable, we show that under standard complexity-theoretic assumptions it does not have a polynomial-size problem kernel when parameterized by the number of vertex deletions. Nevertheless, we develop several effective data reduction and pruning rules, resulting in a competitive solver, outperforming a standard CPLEX solver in most instances of an established biological test data set.

A. Figiel—Partially supported by DFG project NI 369/18.

A.-S. Himmel—Supported by DFG project NI 369/16.

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Notes

  1. 1.

    This is the generalization of Cluster Editing where clusters are requested to be s-plexes (and not cliques); an s-plex is a subgraph where each vertex is connected to all other vertices of the s-plex except for at most \(s-1\) vertices. Notably, a clique is a 1-plex.

  2. 2.

    It has been featured as an open problem whether the edge deletion variant s -Club Cluster Edge Deletion has a polynomial-size problem kernel [7, 25].

  3. 3.

    Given an undirected graph \(G = (V,E)\) and an integer k, the question is whether there is a dominating set \(V' \subseteq V\) (that is, \(N[V'] = V\)) of size at most k.

  4. 4.

    The source code is available at https://fpt.akt.tu-berlin.de/software/two-club-editing/two-club-vertex-deletion.zip and includes the source code for the ILP formulation using CPLEX.

  5. 5.

    The dataset is available at https://bio.informatik.uni-jena.de/data/#cluster_editing_data.

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Acknowledgment

We thank anonymous reviewers for their valuable feedback.

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Authors and Affiliations

  1. TU Berlin, Faculty IV, Algorithmics and Computational Complexity, Berlin, Germany

    Aleksander Figiel, Anne-Sophie Himmel, André Nichterlein & Rolf Niedermeier

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  1. Aleksander Figiel
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Correspondence to André Nichterlein .

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  1. Sapienza University of Rome, Rome, Italy

    Tiziana Calamoneri

  2. Sapienza University of Rome, Rome, Italy

    Federico Corò

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Figiel, A., Himmel, AS., Nichterlein, A., Niedermeier, R. (2021). On 2-Clubs in Graph-Based Data Clustering: Theory and Algorithm Engineering. In: Calamoneri, T., Corò, F. (eds) Algorithms and Complexity. CIAC 2021. Lecture Notes in Computer Science(), vol 12701. Springer, Cham. https://doi.org/10.1007/978-3-030-75242-2_15

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