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Finding and Counting Vertex-Colored Subtrees

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Abstract

The problems studied in this article originate from the Graph Motif problem introduced by Lacroix et al. (IEEE/ACM Trans. Comput. Biol. Bioinform. 3(4):360–368, 2006) in the context of biological networks. The problem is to decide if a vertex-colored graph has a connected subgraph whose colors equal a given multiset of colors M. It is a graph pattern-matching problem variant, where the structure of the occurrence of the pattern is not of interest but the only requirement is the connectedness. Using an algebraic framework recently introduced by Koutis (Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, vol. 5125, pp. 575–586, 2008) and Koutis and Williams (Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, vol. 5555, pp. 653–664, 2009), we obtain new FPT algorithms for Graph Motif and variants, with improved running times. We also obtain results on the counting versions of this problem, proving that the counting problem is FPT if M is a set, but becomes #W[1]-hard if M is a multiset with two colors. Finally, we present an experimental evaluation of this approach on real datasets, showing that its performance compares favorably with existing software.

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Notes

  1. Where μ(t,t′) is 0 if there exists i∈[k] s.t. \(t_{i} - t'_{i} > 1\), and is otherwise equal to (−1)r where r is the number of i∈[k] s.t. \(t_{i} - t'_{i} = 1\).

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Acknowledgements

The authors acknowledge partial funding from DFG PABI BO1910/9-1 and ANR project BIRDS JCJC SIMI 2-2010, and also would like to thanks Sharon Bruckner, Khanh-Lam Mai and the anonymous reviewers for valuable comments and remarks.

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Authors and Affiliations

  1. Lehrstuhl für Bioinformatik, Friedrich-Schiller Universität Jena, Ernst-Abbe Platz 2, 00743, Jena, Germany

    Sylvain Guillemot & Florian Sikora

  2. Department of Computer Science, Iowa State University, Ames, IA, 50011, USA

    Sylvain Guillemot

  3. Université Paris-Est, LIGM–UMR CNRS 8049, 77454, Marne-la-Vallée Cedex 2, France

    Florian Sikora

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  1. Sylvain Guillemot
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  2. Florian Sikora
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Correspondence to Florian Sikora.

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An extended abstract of this paper appeared in MFCS 2010 [15].

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Guillemot, S., Sikora, F. Finding and Counting Vertex-Colored Subtrees. Algorithmica 65, 828–844 (2013). https://doi.org/10.1007/s00453-011-9600-8

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