Algorithmica

, Volume 71, Issue 3, pp 731–757 | Cite as

Incompressibility of \(H\)-Free Edge Modification Problems

Article
First Online:
Received:
Accepted:

Abstract

Given a fixed graph \(H\), the \(H\)-Free Edge Deletion (resp., Completion, Editing) problem asks whether it is possible to delete from (resp., add to, delete from or add to) the input graph at most \(k\) edges so that the resulting graph is \(H\)-free, i.e., contains no induced subgraph isomorphic to \(H\). These \(H\)-free edge modification problems are well known to be fixed-parameter tractable for every fixed \(H\). In this paper we study the incompressibility, i.e., nonexistence of polynomial kernels, for these \(H\)-free edge modification problems in terms of the structure of \(H\), and completely characterize their nonexistence for \(H\) being paths, cycles or 3-connected graphs. We also give a sufficient condition for the nonexistence of polynomial kernels for \({\mathcal {F}}\)-Free Edge Deletion problems, where \({\mathcal {F}}\) is a finite set of forbidden induced subgraphs. As an effective tool, we have introduced an incompressible constraint satisfiability problem Propagational-\(f\) Satisfiability to express common propagational behaviors of events, and we expect the problem to be useful in studying the nonexistence of polynomial kernels in general.

Keywords

Parameterized complexity Polynomial kernel Polynomial compression Incompressibility Edge modification 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The paper is partially based on the M.Phil. Thesis of the 2nd author under the supervision of the 1st author. We thank the two anonymous reviewers for their constructive suggestions.

References

  1. 1.
    Bodlaender, H.L., Cai, L., Chen, J., Fellows, M.R., Telle, J.A., Marx, D.: Open Problems in Parameterized and Exact Computation—IWPEC 2006. Utrecht University Technical Report UU-CS-2006-052 (2006)Google Scholar
  2. 2.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bodlaender, H.L., Jansen, B., Kratsch, S.: Kernelization lower bounds by cross-compositions. SIAM J. Discrete Math. 28(1), 277–305 (2014)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412(35), 4570–4578 (2011)CrossRefMATHGoogle Scholar
  5. 5.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 157–206 (1996)CrossRefGoogle Scholar
  6. 6.
    Cai, L., Cai, Y.: Incompressibility of \(H\)-free edge modification. In: Gutin, G., Szeider, S. (eds.) 8th International Symposium on Parameterized and Exact Computation (IPEC 2013), Lecture Notes in Computer Science, vol. 8246, pp. 84–96 (2013)Google Scholar
  7. 7.
    Cai, Y.: Polynomial Kernelisation of \(H\)-free Edge Modification Problems, MPhil Thesis, Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong SAR, China (2012). http://www.uni-marburg.de/fb12/ps/team/cai-masterarbeit.pdf
  8. 8.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeled data clustering: fixed parameter algorithms for clique generation. Theory Comput. Syst. 38(4), 373–392 (2005)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Guillemot, S., Paul, C., Perez, A.: P On the (non-)existence of polynomial kernels for \(P_l\)-free edge modification problems. In: Raman, V., Saurabh, S. (eds.), 5th International Symposium on Parameterized and Exact Computation (IPEC 2010), Lecture Notes in Computer Science, vol. 6478, pp. 147–157 (2010)Google Scholar
  12. 12.
    Guillemot, S., Havet, F., Paul, C., Perez, A.: On the (non-)existence of polynomial kernels for \(P_l\)-free edge modification problems. Algorithmica 65(4), 900–926 (2013)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Guo, J.: Problem kernels for NP-complete edge deletion problems: split and related graphs. In: Tokuyama, T. (ed.), 18th International Symposium on Algorithms and Complexity (ISAAC 2007), Lecture Notes in Computer Science, vol. 4835, pp. 915–926 (2007)Google Scholar
  14. 14.
    Kratsch, S., Wahlström, M.: Two edge modification problems without polynomial kernels. Discrete Optim. 10, 193–199 (2013)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringThe Chinese University of Hong KongShatinChina
  2. 2.Philipps-UniversitaetMarburgGermany

Personalised recommendations