Parameterizing Edge Modification Problems Above Lower Bounds

  • René van Bevern
  • Vincent Froese
  • Christian Komusiewicz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)

Abstract

For a fixed graph F, we study the parameterized complexity of a variant of the \(F\text {-}{\textsc {free\ Editing}}\) problem: Given a graph G and a natural number k, is it possible to modify at most k edges in G so that the resulting graph contains no induced subgraph isomorphic to F? In our variant, the input additionally contains a vertex-disjoint packing \(\mathcal H\) of induced subgraphs of G, which provides a lower bound \(h(\mathcal H)\) on the number of edge modifications required to transform G into an F-free graph. While earlier works used the number k as parameter or structural parameters of the input graph G, we consider instead the parameter \(\ell :=k-h(\mathcal H)\), that is, the number of edge modifications above the lower bound \(h(\mathcal H)\). We show fixed-parameter tractability with respect to \(\ell \) for \(K_3\text {-}\textsc {Free\ Editing}\), Feedback Arc Set in Tournaments, and Cluster Editing when the packing \(\mathcal H\) contains subgraphs with bounded solution size. For \(K_3\text {-}\textsc {Free\ Editing}\), we also prove NP-hardness in case of edge-disjoint packings of \(K_3\)s and \(\ell =0\), while for \(K_q\text {-}\textsc {Free\ Editing}\) and \(q\ge 6\), NP-hardness for \(\ell =0\) even holds for vertex-disjoint packings of \(K_q\)s.

Keywords

NP-hard problem Fixed-parameter algorithm Subgraph packing Kernelization Graph-based clustering Feedback arc set Cluster editing 

References

  1. 1.
    Alon, N.: Ranking tournaments. SIAM J. Discrete Math. 20(1), 137–142 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aravind, N.R., Sandeep, R.B., Sivadasan, N.: Parameterized lower bounds and dichotomy results for the NP-completeness of H-free edge modification problems. In: Kranakis, E., Navarro, G., Chávez, E. (eds.) LATIN 2016: Theoretical Informatics. LNCS, vol. 9644. Springer, Heidelberg (2016)Google Scholar
  3. 3.
    Bessy, S., Fomin, F.V., Gaspers, S., Paul, C., Perez, A., Saurabh, S., Thomassé, S.: Kernels for feedback arc set in tournaments. J. Comput. Syst. Sci. 77(6), 1071–1078 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    van Bevern, R.: Towards optimal and expressive kernelization for \(d\)-Hitting Set. Algorithmica 70(1), 129–147 (2014)MathSciNetMATHGoogle Scholar
  5. 5.
    Böcker, S.: A golden ratio parameterized algorithm for cluster editing. J. Discrete Algorithms 16, 79–89 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Böcker, S., Briesemeister, S., Bui, Q.B.A., Truß, A.: Going weighted: parameterized algorithms for cluster editing. Theor. Comput. Sci. 410(52), 5467–5480 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brügmann, D., Komusiewicz, C., Moser, H.: On generating triangle-free graphs. In: Proceedings of the DIMAP Workshop on Algorithmic Graph Theory (AGT 2009). pp. 51–58. ENDM, Elsevier (2009)Google Scholar
  8. 8.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chen, J., Meng, J.: A \(2k\) kernel for the cluster editing problem. J. Comput. Syst. Sci. 78(1), 211–220 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Switzerland (2015)CrossRefMATHGoogle Scholar
  11. 11.
    Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: On multiway cut parameterized above lower bounds. ACM T. Comput. Theor. 5(1), 3 (2013)MathSciNetMATHGoogle Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Heidelberg (2013)CrossRefMATHGoogle Scholar
  13. 13.
    Fellows, M.R.: Blow-ups, win/win’s, and crown rules: some new directions in FPT. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 1–12. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Fomin, F.V., Kratsch, S., Pilipczuk, M., Pilipczuk, M., Villanger, Y.: Subexponential fixed-parameter tractability of cluster editing, manuscript available on arXiv. arXiv:1112.4419
  15. 15.
    Garg, S., Philip, G.: Raising the bar for vertex cover: fixed-parameter tractability above a higher guarantee. In: Proceedings of the 27th SODA. SIAM (2016)Google Scholar
  16. 16.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Automated generation of search tree algorithms for hard graph modification problems. Algorithmica 39(4), 321–347 (2004)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hartung, S., Hoos, H.H.: Programming by optimisation meets parameterised algorithmics: a case study for cluster editing. In: Jourdan, L., Dhaenens, C., Marmion, M.-E. (eds.) LION 9 2015. LNCS, vol. 8994, pp. 43–58. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  18. 18.
    Karpinski, M., Schudy, W.: Faster algorithms for feedback arc set tournament, Kemeny rank aggregation and betweenness tournament. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part I. LNCS, vol. 6506, pp. 3–14. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Komusiewicz, C., Uhlmann, J.: Cluster editing with locally bounded modifications. Discrete Appl. Math. 160(15), 2259–2270 (2012)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Křivánek, M., Morávek, J.: NP-hard problems in hierarchical-tree clustering. Acta Informatica 23(3), 311–323 (1986)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    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
  23. 23.
    Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31(2), 335–354 (1999)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Moser, H., Niedermeier, R., Sorge, M.: Exact combinatorial algorithms and experiments for finding maximum \(k\)-plexes. J. Comb. Optim. 24(3), 347–373 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Paul, C., Perez, A., Thomassé, S.: Conflict packing yields linear vertex-kernels for \(k\)-FAST, \(k\)-dense RTI and a related problem. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 497–507. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  26. 26.
    Razgon, I., O’Sullivan, B.: Almost 2-SAT is fixed-parameter tractable. J. Comput. Syst. Sci. 75(8), 435–450 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144(1–2), 173–182 (2004)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wahlström, M.: Algorithms, measures and upper bounds for satisfiability and related problems. Ph.D. thesis, Linköpings universitet (2007)Google Scholar
  29. 29.
    Yannakakis, M.: Edge-deletion problems. SIAM J. Comput. 10(2), 297–309 (1981)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • René van Bevern
    • 1
  • Vincent Froese
    • 2
  • Christian Komusiewicz
    • 3
  1. 1.Novosibirsk State UniversityNovosibirskRussian Federation
  2. 2.Technische Universität BerlinBerlinGermany
  3. 3.Friedrich-Schiller-Universität JenaJenaGermany

Personalised recommendations