The Parameterized Complexity of Cycle Packing: Indifference is Not an Issue

  • R. Krithika
  • Abhishek Sahu
  • Saket Saurabh
  • Meirav Zehavi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

In the Cycle Packing problem, we are given an undirected graph G, a positive integer r, and the task is to check whether there exist r vertex-disjoint cycles. In this paper, we study Cycle Packing with respect to a structural parameter, namely, distance to proper interval graphs (indifference graphs). In particular, we show that Cycle Packing is fixed-parameter tractable (FPT) when parameterized by t, the size of a proper interval deletion set. For this purpose, we design an algorithm with \(\mathcal {O}(2^{\mathcal {O}(t \log t)} n^{\mathcal {O}(1)})\) running time. Several structural parameterizations for Cycle Packing have been studied in the literature and our FPT algorithm fills a gap in the ecology of such parameterizations. We combine color coding, greedy strategy and dynamic programming based on structural properties of proper interval graphs in a non-trivial fashion to obtain the FPT algorithm.

References

  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L.: On disjoint cycles. Int. J. Found. Comput. Sci. 5(1), 59–68 (1994)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bodlaender, H.L., Jansen, B.M.P.: Vertex cover kernelization revisited: upper and lower bounds for a refined parameter. Theory Comput. Syst. 63(2), 263–299 (2013)MathSciNetMATHGoogle Scholar
  4. 4.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernel bounds for path and cycle problems. Theor. Comput. Sci. 511, 117–136 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412(35), 4570–4578 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cai, L.: Parameterized complexity of vertex colouring. Discrete Appl. Math. 127(3), 415–429 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cao, Y.: Unit interval editing is fixed-parameter tractable. Inf. Comput. 253(Part 1), 109–126 (2017)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cao, Y., Marx, D.: Interval deletion is fixed-parameter tractable. ACM Trans. Algorithms 11(3), 21:1–21:35 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-319-21275-3 CrossRefMATHGoogle Scholar
  10. 10.
    Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 150–159 (2011)Google Scholar
  11. 11.
    Diestel, R.: Graph Theory. GTM, vol. 173. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-662-53622-3 MATHGoogle Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, London (2013).  https://doi.org/10.1007/978-1-4471-5559-1 CrossRefMATHGoogle Scholar
  13. 13.
    Erdös, P., Pósa, L.: On independent circuits contained in a graph. Can. J. Math. 17, 347–352 (1965)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fellows, M.R., Langston, M.A.: Nonconstructive tools for proving polynomial-time decidability. J. ACM 35(3), 727–739 (1988)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fellows, M.R., Lokshtanov, D., Misra, N., Mnich, M., Rosamond, F.A., Saurabh, S.: The complexity ecology of parameters: an illustration using bounded max leaf number. Theory Comput. Syst. 45(4), 822–848 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006).  https://doi.org/10.1007/3-540-29953-X MATHGoogle Scholar
  17. 17.
    Fomin, F.V., Saurabh, S., Villanger, Y.: A polynomial kernel for proper interval vertex deletion. SIAM J. Discrete Math. 27(4), 1964–1976 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57. North-Holland Publishing Co., Amsterdam (2004)MATHGoogle Scholar
  19. 19.
    Guruswami, V., Pandu Rangan, C., Chang, M.S., Chang, G.J., Wong, C.K.: The \(K_r\)-packing problem. Computing 66(1), 79–89 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gutin, G., Kim, E.J., Lampis, M., Mitsou, V.: Vertex cover problem parameterized above and below tight bounds. Theory Comput. Syst. 48(2), 402–410 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Gutin, G., Yeo, A.: Constraint satisfaction problems parameterized above or below tight bounds: a survey. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) The Multivariate Algorithmic Revolution and Beyond. LNCS, vol. 7370, pp. 257–286. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-30891-8_14 CrossRefGoogle Scholar
  22. 22.
    Jansen, B.M.P.: The power of data reduction: kernels for fundamental graph problems. Ph.D. thesis, Utrecht University, The Netherlands (2013)Google Scholar
  23. 23.
    Jansen, B.M.P., Fellows, M.R., Rosamond, F.A.: Towards fully multivariate algorithmics: parameter ecology and the deconstruction of computational complexity. Eur. J. Comb. 34(3), 541–566 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Jansen, B.M.P., Raman, V., Vatshelle, M.: Parameter ecology for feedback vertex set. Tsinghua Sci. Technol. 19(4), 387–409 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ke, Y., Cao, Y., Ouyang, X., Wang, J.: Unit interval vertex deletion: fewer vertices are relevant (2016). arXiv:arXiv:1607.01162
  26. 26.
    Kloks, T.: Packing interval graphs with vertex-disjoint triangles. CoRR, abs/1202.1041 (2012)Google Scholar
  27. 27.
    Lokshtanov, D., Mouawad, A., Saurabh, S., Zehavi, M.: Packing cycles faster than Erdös-Pósa. In: 44th International Colloquium on Automata, Languages, and Programming (ICALP), pp. 71:1–71:15 (2017)Google Scholar
  28. 28.
    Lokshtanov, D., Narayanaswamy, N.S., Raman, V., Ramanujan, M.S., Saurabh, S.: Faster parameterized algorithms using linear programming. ACM Trans. Algorithms 11(2), 15:1–15:31 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lokshtanov, D., Panolan, F., Ramanujan, M.S., Saurabh, S.: Lossy kernelization. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pp. 224–237 (2017)Google Scholar
  30. 30.
    Looges, P.J., Olariu, S.: Optimal greedy algorithms for indifference graphs. Comput. Math. Appli. 25(7), 15–25 (1993)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Manić, G., Wakabayashi, Y.: Packing triangles in low degree graphs and indifference graphs. Discrete Math. 308(8), 1455–1471 (2008)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: IEEE 36th Annual Symposium on Foundations of Computer Science (FOCS), pp. 182–191 (1995)Google Scholar
  33. 33.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B 63(1), 65–110 (1995)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    van Bevern, R., Komusiewicz, C., Moser, H., Niedermeier, R.: Measuring indifference: unit interval vertex deletion. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 232–243. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-16926-7_22 CrossRefGoogle Scholar
  35. 35.
    van’t Hof, P., Villanger, Y.: Proper interval vertex deletion. Algorithmica 65(4), 845–867 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • R. Krithika
    • 1
    • 2
  • Abhishek Sahu
    • 1
    • 2
  • Saket Saurabh
    • 1
    • 2
    • 3
  • Meirav Zehavi
    • 4
  1. 1.The Institute of Mathematical Sciences, HBNIChennaiIndia
  2. 2.UMI ReLaxChennaiIndia
  3. 3.University of BergenBergenNorway
  4. 4.Ben-Gurion UniversityBeershebaIsrael

Personalised recommendations