Skip to main content
SpringerLink
Log in

Parameterized Algorithms for Directed Modular Width

  • Conference paper
  • First Online:
Algorithms and Discrete Applied Mathematics (CALDAM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12016))

Included in the following conference series:

Abstract

Many well-known \(\mathsf {NP}\)-hard algorithmic problems on directed graphs resist efficient parameterizations with most known width measures for directed graphs, such as directed treewidth, DAG-width, Kelly-width and many others. While these focus on measuring how close a digraph is to an oriented tree resp. a directed acyclic graph, in this paper, we investigate directed modular width as a parameter, which is closer to the concept of clique-width. We investigate applications of modular decompositions of directed graphs to a wide range of algorithmic problems and derive FPT algorithms for several well-known digraph-specific \(\mathsf {NP}\)-hard problems, namely minimum (weight) directed feedback vertex set, minimum (weight) directed dominating set, digraph colouring, directed Hamiltonian path/cycle, partitioning into paths, (capacitated) vertex-disjoint directed paths, and the directed subgraph homeomorphism problem. The latter yields a polynomial-time algorithm for detecting topological minors in digraphs of bounded directed modular width. Finally we illustrate that other structural digraph parameters, such as directed pathwidth and cycle-rank can be computed efficiently using directed modular width as a parameter.

A full version of this article is available at https://arxiv.org/abs/1905.13203.

R. Steiner—Supported by DFG-GRK 2434.

S. Wiederrecht—Supported by the ERC consolidator grant DISTRUCT-648527.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    https://arxiv.org/abs/1905.13203.

  2. 2.

    Reachability in directed graphs is a simple special case of the max-flow problem and can be solved using one of the well-known polynomial algorithms for this task (see for instance [12]).

References

  1. Bang-Jensen, J., Gutin, G.: Classes of Directed Graphs. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-71840-8

    Book  MATH  Google Scholar 

  2. Bang-Jensen, J., Havet, F., Trotignon, N.: Finding an induced subdivision of a digraph. Theor. Comput. Sci. 443, 10–24 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S.: DAG-width and parity games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 524–536. Springer, Heidelberg (2006). https://doi.org/10.1007/11672142_43

    Chapter  Google Scholar 

  4. Bodlaender, H.L.: Treewidth: algorithmic techniques and results. In: Prívara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 19–36. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0029946

    Chapter  Google Scholar 

  5. Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55(5), 21 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Courcelle, B.: The monadic second-order logic of graphs. i. recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Courcelle, B.: The expression of graph properties and graph transformations in monadic second-order logic. In: Handbook of Graph Grammars and Computing By Graph Transformation: Volume 1: Foundations, pp. 313–400. World Scientific (1997)

    Google Scholar 

  8. Courcelle, B., Engelfriet, J.: Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, vol. 138. Cambridge University Press, New York (2012)

    Book  MATH  Google Scholar 

  9. Courcelle, B., Engelfriet, J., Rozenberg, G.: Handle-rewriting hypergraph grammars. J. Comput. Syst. Sci. 46(2), 218–270 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cygan, M., Marx, D., Pilipczuk, M.: The planar directed k-vertex-disjoint paths problem is fixed-parameter tractable. In: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pp. 197–206 (2013)

    Google Scholar 

  12. Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19, 248–264 (1972)

    Article  MATH  Google Scholar 

  13. Eggan, L.C., et al.: Transition graphs and the star-height of regular events. The Mich. Math. J. 10(4), 385–397 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  14. Espelage, W., Gurski, F., Wanke, E.: How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 117–128. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45477-2_12

    Chapter  MATH  Google Scholar 

  15. Fellows, M.R., Lokshtanov, D., Misra, N., Rosamond, F.A., Saurabh, S.: Graph layout problems parameterized by vertex cover. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 294–305. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-92182-0_28

    Chapter  Google Scholar 

  16. Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Intractability of clique-width parameterizations. SIAM J. Comput. 39(5), 1941–1956 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theor. Comput. Sci. 10, 111–121 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  18. Frank, A.: Packing paths, circuits and cuts-a survey. In: Paths, Flows, and VLSI-Layout, pp. 47–100. Springer-Verlag, Berlin (1990)

    Google Scholar 

  19. Gajarský, J., Lampis, M., Ordyniak, S.: Parameterized algorithms for modular-width. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 163–176. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-03898-8_15

    Chapter  MATH  Google Scholar 

  20. Ganian, R., et al.: Are there any good digraph width measures? J. Comb. Theory Ser. B 116, 250–286 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ganian, R., Hliněný, P., Kneis, J., Langer, A., Obdržálek, J., Rossmanith, P.: Digraph width measures in parameterized algorithmics. Discrete Appl. Math. 168, 88–107 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1990)

    MATH  Google Scholar 

  23. Giannopoulou, A.C., Hunter, P., Thilikos, D.M.: Lifo-search: a min-max theorem and a searching game for cycle-rank and tree-depth. Discrete Appl. Math. 160(15), 2089–2097 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Grohe, M., Kawarabayashi, K.I., Marx, D., Wollan, P.: Finding topological subgraphs is fixed-parameter tractable. In: Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC 2011, New York, NY, USA, pp. 479–488 (2011)

    Google Scholar 

  25. Gurski, F., Komander, D., Rehs, C.: Computing digraph width measures on directed co-graphs. In: Gąsieniec, L.A., Jansson, J., Levcopoulos, C. (eds.) FCT 2019. LNCS, vol. 11651, pp. 292–305. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-25027-0_20

    Chapter  Google Scholar 

  26. Hunter, P., Kreutzer, S.: Digraph measures: kelly decompositions, games, and orderings. Theor. Comput. Sci. 399(3), 206–219 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Johnson, T., Robertson, N., Seymour, P., Thomas, R.: Directed tree-width. J. Comb. Theory Ser. B 82(1), 138–154 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. McConnell, R.M., Spinrad, J.P.: Modular decomposition and transitive orientation. Discrete Math. 201(1–3), 189–241 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. McConnella, R.M., de Montgolfier, F.: Linear time modular decomposition of directed graphs. Discrete Appl. Math. 145, 198–209 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Millani, M.G., Steiner, R., Wiederrecht, S.: Colouring non-even digraphs. Technical report (2019, submitted). arXiv Preprint, arXiv:1903.02872

  31. N. Robertson, P.D.S.: An outline of a disjoint paths algorithm. In: Paths, Flows, and VLSI-Layout, pp. 267–292. Springer-Verlag, Berlin (1990)

    Google Scholar 

  32. Plesník, J.: The NP-completeness of the Hamiltonian cycle problem in planar digraphs with degree bound two. Inform. Process. Lett. 8(4), 199–201 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  33. Robertson, N., Seymour, P.D.: Graph minors. ii. algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technische Universität Berlin, Berlin, Germany

    Raphael Steiner & Sebastian Wiederrecht

Authors
  1. Raphael Steiner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sebastian Wiederrecht
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sebastian Wiederrecht .

Editor information

Editors and Affiliations

  1. University of Kerala, Thiruvananthapuram, India

    Manoj Changat

  2. Indian Statistical Institute, Kolkata, India

    Sandip Das

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Steiner, R., Wiederrecht, S. (2020). Parameterized Algorithms for Directed Modular Width. In: Changat, M., Das, S. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2020. Lecture Notes in Computer Science(), vol 12016. Springer, Cham. https://doi.org/10.1007/978-3-030-39219-2_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-39219-2_33

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-39218-5

  • Online ISBN: 978-3-030-39219-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation