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Exact Exponential Algorithms to Find a Tropical Connected Set of Minimum Size

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Parameterized and Exact Computation (IPEC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8894))

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Abstract

The input of the Tropical Connected Set problem is a vertex-colored graph \(G=(V,E)\) and the task is to find a connected subset \(S\subseteq V\) of minimum size such that each color of \(G\) appears in \(S\). This problem is known to be NP-complete, even when restricted to trees of height at most three. We show that Tropical Connected Set on trees has no subexponential-time algorithm unless the Exponential Time Hypothesis fails. This motivates the study of exact exponential algorithms to solve Tropical Connected Set. We present an \(\mathcal {O}^*(1.5359^n)\) time algorithm for general graphs and an \(\mathcal {O}^*(1.2721^n)\) time algorithm for trees.

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References

  1. Abu-Khzam, F.N., Mouawad, A.E., Liedloff, M.: An exact algorithm for connected red-blue dominating set. J. Discret. Algorithm. 9(3), 252–262 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ambalath, A.M., Balasundaram, R., Rao H., C., Koppula, V., Misra, N., Philip, G., Ramanujan, M.S.: On the kernelization complexity of colorful motifs. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 14–25. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  3. Angles D’Auriac, J.-A., Cohen, N., El Maftouhi, A., Harutyunyan, A., Legay, S., Manoussakis, Y.: Connected tropical subgraphs in vertex-colored graphs. http://people.maths.ox.ac.uk/harutyunyan/Tropical%20sets.pdf

  4. Betzler, N., Van Bevern, R., Fellows, M.R., Komusiewicz, C., Niedermeier, R.: Parameterized algorithmics for finding connected motifs in biological networks. IEEE/ACM Trans. Comput. Biol. Bioinf. 8(5), 1296–1308 (2011)

    Article  Google Scholar 

  5. Bruckner, S., Hüffner, F., Karp, R.M., Shamir, R., Sharan, R.: Topology-free querying of protein interaction networks. J. Comput. Biol. 17(3), 237–252 (2010)

    Article  MathSciNet  Google Scholar 

  6. Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85, 12–75 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Demaine, E.D., Hajiaghayi, M.T.: The bidimensionality theory and its algorithmic applications. Comput. J. 51, 292–302 (2008)

    Article  Google Scholar 

  8. Dondi, R., Fertin, G., Vialette, S.: Complexity issues in vertex-colored graph pattern matching. J. Discret. Algorithm. 9(1), 82–99 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fellows, M.R., Fertin, G., Hermelin, D., Vialette, S.: Sharp tractability borderlines for finding connected motifs in vertex-colored graphs. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 340–351. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  10. Fellows, M.R., Fertin, G., Hermelin, D., Vialette, S.: Upper and lower bounds for finding connected motifs in vertex-colored graphs. J. Comput. Syst. Sci. 77(4), 799–811 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer, Heidelberg (2010)

    Book  MATH  Google Scholar 

  12. Fomin, F.V., Kratsch, D., Woeginger, G.J.: Exact (Exponential) algorithms for the dominating set problem. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 245–256. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  13. Fomin, F.V., Thilikos, D.M.: A simple and fast approach for solving problems on planar graphs. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 56–67. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  14. Guillemot, S., Sikora, F.: Finding and counting vertex-colored subtrees. Algorithmica 65(4), 828–844 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput/Syst. Sci. 62, 367–375 (2001)

    MATH  MathSciNet  Google Scholar 

  16. Lacroix, V., Fernandes, C.G., Sagot, M.-F.: Motif search in graphs: application to metabolic networks. IEEE/ACM Trans. Comput. Biol. Bioinf. 3(4), 360–368 (2006)

    Article  Google Scholar 

  17. McMorris, F., Warnow, T., Wimer, T.: Triangulating vertex-colored graphs. SIAM J. Discret. Math. 7, 296–306 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nederlof, J.: Fast polynomial-space algorithms using Inclusion-Exclusion. Algorithmica 65, 868–884 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Scott, J., Ideker, T., Karp, R.M., Sharan, R.: Efficient algorithms for detecting signaling pathways in protein interaction networks. J. Comput. Biol. 13, 133–144 (2006)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Université Libre de Bruxelles, CP 212, 1050, Bruxelles, Belgium

    Mathieu Chapelle

  2. LITA, Université de Lorraine, 57045, Metz Cedex 01, France

    Manfred Cochefert & Dieter Kratsch

  3. LIFO, Université d’Orléans, 45067, Orléans Cedex 2, France

    Romain Letourneur & Mathieu Liedloff

Authors
  1. Mathieu Chapelle
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  2. Manfred Cochefert
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  3. Dieter Kratsch
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  4. Romain Letourneur
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  5. Mathieu Liedloff
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Corresponding author

Correspondence to Dieter Kratsch .

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Editors and Affiliations

  1. Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

    Marek Cygan

  2. University of Bergen, Bergen, Norway

    Pinar Heggernes

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Chapelle, M., Cochefert, M., Kratsch, D., Letourneur, R., Liedloff, M. (2014). Exact Exponential Algorithms to Find a Tropical Connected Set of Minimum Size. In: Cygan, M., Heggernes, P. (eds) Parameterized and Exact Computation. IPEC 2014. Lecture Notes in Computer Science(), vol 8894. Springer, Cham. https://doi.org/10.1007/978-3-319-13524-3_13

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  • DOI: https://doi.org/10.1007/978-3-319-13524-3_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-13523-6

  • Online ISBN: 978-3-319-13524-3

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