Skip to main content

Parameterized Complexity of Path Set Packing

  • Conference paper
  • First Online:
WALCOM: Algorithms and Computation (WALCOM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13973))

Included in the following conference series:

Abstract

In PATH SET PACKING, the input is an undirected graph G, a collection \(\mathcal{P}\) of simple paths in G, and a positive integer k. The problem is to decide whether there exist k edge-disjoint paths in \(\mathcal{P}\). We study the parameterized complexity of PATH SET PACKING with respect to both natural and structural parameters. We show that the problem is W[1]-hard with respect to vertex cover plus the maximum length of a path in \(\mathcal{P}\), and W[1]-hard with respect to pathwidth plus maximum degree plus solution size. These results answer an open question raised in [17]. On the positive side, we present an FPT algorithm parameterized by feedback vertex set plus maximum degree, and also provide an FPT algorithm parameterized by treewidth plus maximum degree plus maximum length of a path in \(\mathcal{P}\).

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    The proofs of statements marked with a \(\star \) have been omitted.

References

  1. Abu-Khzam, F.N.: An improved kernelization algorithm for r-set packing. Inf. Process. Lett. 110(16), 621–624 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnborg, S., Proskurowski, A.: Linear time algorithms for np-hard problems restricted to partial k-trees. Discret. Appl. Math. 23(1), 11–24 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bessy, S., Bougeret, M., Chaplick, S., Gonçalves, D., Paul, C.: On independent set in B1-EPG graphs. Discret. Appl. Math. 278, 62–72 (2020)

    Article  MATH  Google Scholar 

  4. Cygan, M., et al.: Parameterized Algorithms. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-319-21275-3

    Book  MATH  Google Scholar 

  5. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  6. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science, Springer, New York (1999). https://doi.org/10.1007/978-1-4612-0515-9

    Book  MATH  Google Scholar 

  7. Epstein, D., Golumbic, M.C., Morgenstern, G.: Approximation algorithms for \(B_{1}\)-EPG graphs. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 328–340. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40104-6_29

    Chapter  MATH  Google Scholar 

  8. Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410(1), 53–61 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Garey, M.R., Johnson, D.S.: The rectilinear steiner tree problem is \$np\$-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  10. Golumbic, M.C., Jamison, R.E.: Edge and vertex intersection of paths in a tree. Discret. Math. 55(2), 151–159 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Golumbic, M.C., Jamison, R.E.: The edge intersection graphs of paths in a tree. J. Comb. Theory Ser. B 38(1), 8–22 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  12. Golumbic, M.C., Lipshteyn, M., Stern, M.: Edge intersection graphs of single bend paths on a grid. Networks 54(3), 130–138 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jansen, B.M.P.: On structural parameterizations of hitting set: hitting paths in graphs using 2-SAT. In: Mayr, E.W. (ed.) WG 2015. LNCS, vol. 9224, pp. 472–486. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53174-7_33

    Chapter  Google Scholar 

  14. Jia, W., Zhang, C., Chen, J.: An efficient parameterized algorithm for m-set packing. J. Algorithms 50(1), 106–117 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Koutis, I.: A faster parameterized algorithm for set packing. Inf. Process. Lett. 94(1), 7–9 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tarjan, R.E.: Decomposition by clique separators. Discret. Math. 55(2), 221–232 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Xu, C., Zhang, G.: The path set packing problem. In: Wang, L., Zhu, D. (eds.) COCOON 2018. LNCS, vol. 10976, pp. 305–315. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94776-1_26

    Chapter  Google Scholar 

Download references

Acknowledgements

We thank anonymous reviewers of this and an earlier version of this paper for useful suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Roopam Saxena .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Aravind, N.R., Saxena, R. (2023). Parameterized Complexity of Path Set Packing. In: Lin, CC., Lin, B.M.T., Liotta, G. (eds) WALCOM: Algorithms and Computation. WALCOM 2023. Lecture Notes in Computer Science, vol 13973. Springer, Cham. https://doi.org/10.1007/978-3-031-27051-2_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-27051-2_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-27050-5

  • Online ISBN: 978-3-031-27051-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics