A Subexponential Parameterized Algorithm for Proper Interval Completion

  • Ivan Bliznets
  • Fedor V. Fomin
  • Marcin Pilipczuk
  • Michał Pilipczuk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8737)

Abstract

In the Proper Interval Completion problem we are given a graph G and an integer k, and the task is to turn G using at most k edge additions into a proper interval graph, i.e., a graph admitting an intersection model of equal-length intervals on a line. The study of Proper Interval Completion from the viewpoint of parameterized complexity has been initiated by Kaplan, Shamir and Tarjan [FOCS 1994; SIAM J. Comput. 1999], who showed an algorithm for the problem working in \(\mathcal{O}(16^k\cdot (n+m))\) time. In this paper we present an algorithm with running time \(k^{\mathcal{O}(k^{2/3})} + \mathcal{O}(nm(kn+m))\), which is the first subexponential parameterized algorithm for Proper Interval Completion.

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References

  1. 1.
    Alon, N., Lokshtanov, D., Saurabh, S.: Fast FAST. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 49–58. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Bessy, S., Perez, A.: Polynomial kernels for Proper Interval Completion and related problems. Information and Computation 231, 89 (2013)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bliznets, I., Fomin, F.V., Pilipczuk, M., Pilipczuk, M.: A subexponential parameterized algorithm for interval completion. CoRR abs/1402.3473 (2014)Google Scholar
  4. 4.
    Bliznets, I., Fomin, F.V., Pilipczuk, M., Pilipczuk, M.: A subexponential parameterized algorithm for proper interval completion. CoRR abs/1402.3472 (2014)Google Scholar
  5. 5.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)CrossRefMATHGoogle Scholar
  6. 6.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs. J. ACM 52(6), 866–893 (2005)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Drange, P.G., Fomin, F.V., Pilipczuk, M., Villanger, Y.: Exploring subexponential parameterized complexity of completion problems. In: STACS 2014 (2014)Google Scholar
  8. 8.
    Feige, U.: Coping with the NP-hardness of the graph bandwidth problem. In: Halldórsson, M.M. (ed.) SWAT 2000. LNCS, vol. 1851, pp. 10–19. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Fomin, F.V., Villanger, Y.: Subexponential parameterized algorithm for minimum fill-in. SIAM J. Comput. 42(6), 2197–2216 (2013)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Ghosh, E., Kolay, S., Kumar, M., Misra, P., Panolan, F., Rai, A., Ramanujan, M.S.: Faster parameterized algorithms for deletion to split graphs. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 107–118. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Kaplan, H., Shamir, R., Tarjan, R.E.: Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput. 28(5), 1906–1922 (1999)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Kratsch, S., Wahlström, M.: Two edge modification problems without polynomial kernels. Discrete Optimization 10(3), 193–199 (2013)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Liu, Y., Wang, J., Xu, C., Guo, J., Chen, J.: An effective branching strategy for some parameterized edge modification problems with multiple forbidden induced subgraphs. In: Du, D.-Z., Zhang, G. (eds.) COCOON 2013. LNCS, vol. 7936, pp. 555–566. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  15. 15.
    Looges, P.J., Olariu, S.: Optimal greedy algorithms for indifference graphs. Computers and Mathematics with Applications 25(7), 15–25 (1993)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Villanger, Y., Heggernes, P., Paul, C., Telle, J.A.: Interval completion is fixed parameter tractable. SIAM J. Comput. 38(5), 2007–2020 (2009)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM J. Alg. Disc. Meth. 2, 77–79 (1981)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Ivan Bliznets
    • 1
  • Fedor V. Fomin
    • 1
    • 2
  • Marcin Pilipczuk
    • 2
  • Michał Pilipczuk
    • 2
  1. 1.St. Petersburg Department of SteklovInstitute of MathematicsRussia
  2. 2.Department of InformaticsUniversity of BergenNorway

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