, Volume 81, Issue 9, pp 3803–3841 | Cite as

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

  • R. KrithikaEmail author
  • Abhishek Sahu
  • Saket Saurabh
  • Meirav Zehavi
Original Research


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. Bodlaender et al. (Theor Comput Sci 511:117–136, 2013) studied several structural parameterizations for Cycle Packing and our FPT algorithm fills a gap in their ecology of 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. Our belief is that this approach is quite general and can be useful in solving many other problems with the same parameterization.


Cycle packing Proper interval deletion set Fixed-parameter tractable 



  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L.: On disjoint cycles. Int. J. Found. Comput. Sci. 5(1), 59–68 (1994)CrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cai, L.: Parameterized complexity of vertex colouring. Discrete Appl. Math. 127(3), 415–429 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cao, Y.: Unit interval editing is fixed-parameter tractable. Inf. Comput. 253, 109–126 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cao, Y., Marx, D.: Interval deletion is fixed-parameter tractable. ACM Trans. Algorithms 11(3), 21:1–21:35 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)CrossRefzbMATHGoogle 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. Springer, Berlin (2006)zbMATHGoogle Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, London (2013)CrossRefzbMATHGoogle Scholar
  13. 13.
    Erdös, P., Pósa, L.: On independent circuits contained in a graph. Can. J. Math. 17, 347–352 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fellows, M.R., Langston, M.A.: Nonconstructive tools for proving polynomial-time decidability. J. ACM 35(3), 727–739 (1988)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Golumbic, M.C.: Algorithmic Graph Theory for Perfect Graphs. Springer, Berlin (2004)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gutin, G., Yeo, A.: Constraint satisfaction problems parameterized above or below tight bounds: a Survey. In: The Multivariate Algorithmic Revolution and Beyond: Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, pp. 257–286 (2012)Google 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)MathSciNetCrossRefzbMATHGoogle 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. arXiv:1607.01162 (2016)
  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. To appear in ICALP (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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lokshtanov, D., Panolan, F., Sridharan, R., Saurabh, S.: Lossy kernelization. To appear in STOC (2017)Google Scholar
  30. 30.
    Looges, P.J., Olariu, S.: Optimal greedy algorithms for indifference graphs. Comput. Math. Appl. 25(7), 15–25 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Manić, G., Wakabayashi, Y.: Packing triangles in low degree graphs and indifference graphs. Discrete Math. 308(8), 1455–1471 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    McKee, T., McMorris, F.: Topics in Intersection Graph Theory. Society for Industrial and Applied Mathematics. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  33. 33.
    Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: Proceedings of IEEE 36th Annual Foundations of Computer Science, pp. 182–191 (1995)Google Scholar
  34. 34.
    van Bevern, R., Komusiewicz, C., Moser, H., Niedermeier, R.: Measuring indifference: unit interval vertex deletion. In: Proceedings of the 36th International Conference on Graph-Theoretic Concepts in Computer Science, WG’10, pp. 232–243. Springer (2010)Google Scholar
  35. 35.
    van’t Hof, P., Villanger, Y.: Proper interval vertex deletion. Algorithmica 65(4), 845–867 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Roberts, F.S.: Indifference graphs. In: Harary, F. (ed.) Proof Techniques in Graph Theory, pp. 139–146. Academic Press, New York (1969)Google Scholar
  37. 37.
    Roberts, F.S.: Indifference and seriation. Ann. N. Y. Acad. Sci. 328(1), 173–182 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B 63(1), 65–110 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

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

Personalised recommendations