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On the Maximum Colorful Arborescence Problem and Color Hierarchy Graph Structure

Authors Guillaume Fertin , Julien Fradin, Christian Komusiewicz



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Author Details

Guillaume Fertin
  • LS2N UMR CNRS 6004, Université de Nantes, Nantes, France
Julien Fradin
  • LS2N UMR CNRS 6004, Université de Nantes, Nantes, France
Christian Komusiewicz
  • Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

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Guillaume Fertin, Julien Fradin, and Christian Komusiewicz. On the Maximum Colorful Arborescence Problem and Color Hierarchy Graph Structure. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 17:1-17:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.CPM.2018.17

Abstract

Let G=(V,A) be a vertex-colored arc-weighted directed acyclic graph (DAG) rooted in some vertex r. The color hierarchy graph H(G) of G is defined as follows: the vertex set of H(G) is the color set C of G, and H(G) has an arc from c to c' if G has an arc from a vertex of color c to a vertex of color c'. We study the Maximum Colorful Arborescence (MCA) problem, which takes as input a DAG G such that H(G) is also a DAG, and aims at finding in G a maximum-weight arborescence rooted in r in which no color appears more than once. The MCA problem models the de novo inference of unknown metabolites by mass spectrometry experiments. Although the problem has been introduced ten years ago (under a different name), it was only recently pointed out that a crucial additional property in the problem definition was missing: by essence, H(G) must be a DAG. In this paper, we further investigate MCA under this new light and provide new algorithmic results for this problem, with a focus on fixed-parameter tractability (FPT) issues for different structural parameters of H(G). In particular, we develop an O^*(3^{{x_H}})-time algorithm for solving MCA, where {x_{H}} is the number of vertices of indegree at least two in H(G), thereby improving the O^*(3^{|C|})-time algorithm of Böcker et al. [Proc. ECCB '08]. We also prove that MCA is W[2]-hard with respect to the treewidth t_H of the underlying undirected graph of H(G), and further show that it is FPT with respect to t_H + l_{C}, where l_{C} := |V| - |C|.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Dynamic programming
Keywords
  • Subgraph problem
  • computational complexity
  • algorithms
  • fixed-parameter tractability
  • kernelization

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References

  1. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, 1995. Google Scholar
  2. Nadja Betzler, René van Bevern, Michael R. Fellows, Christian Komusiewicz, and Rolf Niedermeier. Parameterized algorithmics for finding connected motifs in biological networks. IEEE/ACM Trans. Comput. Biology Bioinform., 8(5):1296-1308, 2011. Google Scholar
  3. Andreas Björklund, Petteri Kaski, and Lukasz Kowalik. Constrained multilinear detection and generalized graph motifs. Algorithmica, 74(2):947-967, 2016. Google Scholar
  4. Sebastian Böcker and Florian Rasche. Towards de novo identification of metabolites by analyzing tandem mass spectra. In Proceedings of the 7th European Conference on Computational Biology (ECCB '08), volume 24(16), pages i49-i55, 2008. Google Scholar
  5. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009. URL: http://dx.doi.org/10.1016/j.jcss.2009.04.001.
  6. Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci., 412(35):4570-4578, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2011.04.039.
  7. Sharon Bruckner, Falk Hüffner, Richard M. Karp, Ron Shamir, and Roded Sharan. Topology-free querying of protein interaction networks. J. Comput. Biol., 17(3):237-252, 2010. Google Scholar
  8. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  9. Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. Kernelization hardness of connectivity problems in d-degenerate graphs. Discr. Appl. Math., 160(15):2131-2141, 2012. Google Scholar
  10. Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Kernelization lower bounds through colors and ids. ACM Trans. Algorithms, 11(2):13:1-13:20, 2014. URL: http://dx.doi.org/10.1145/2650261.
  11. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: http://dx.doi.org/10.1007/978-1-4471-5559-1.
  12. Kai Dührkop, Marie Anne Lataretu, W. Timothy J. White, and Sebastian Böcker. Heuristic algorithms for the maximum colorful subtree problem. arXiv, 2018. URL: https://arxiv.org/abs/1801.07456.
  13. Michael R. Fellows, Guillaume Fertin, Danny Hermelin, and Stéphane Vialette. Upper and lower bounds for finding connected motifs in vertex-colored graphs. J. Comput. Syst. Sci., 77(4):799-811, 2011. Google Scholar
  14. Guillaume Fertin, Julien Fradin, and Géraldine Jean. Algorithmic aspects of the maximum colorful arborescence problem. In T. V. Gopal, Gerhard Jäger, and Silvia Steila, editors, Theory and Applications of Models of Computation - 14th Annual Conference, TAMC 2017, Bern, Switzerland, April 20-22, 2017, Proceedings, volume 10185 of Lecture Notes in Computer Science, pages 216-230, 2017. URL: http://dx.doi.org/10.1007/978-3-319-55911-7_16.
  15. Guillaume Fertin and Christian Komusiewicz. Graph motif problems parameterized by dual. In Proceedings of the 27th Annual Symposium on Combinatorial Pattern Matching (CPM '16), volume 54 of LIPIcs, pages 7:1-7:12. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  16. Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: Distance from triviality. In Proceedings of the First International Workshop on Parameterized and Exact Computation (IWPEC '04), volume 3162 of LNCS, pages 162-173. Springer, 2004. Google Scholar
  17. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York., The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: http://www.cs.berkeley.edu/~luca/cs172/karp.pdf.
  18. Vincent Lacroix, Cristina G. Fernandes, and Marie-France Sagot. Motif search in graphs: Application to metabolic networks. IEEE/ACM Trans. Comput. Biology Bioinform., 3(4):360-368, 2006. Google Scholar
  19. Imran Rauf, Florian Rasche, Francois Nicolas, and Sebastian Böcker. Finding maximum colorful subtrees in practice. J. Comput. Biol., 20(4):311-321, 2013. Google Scholar
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