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On the (Non-)Existence of Polynomial Kernels for P l -Free Edge Modification Problems

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Given a graph G=(V,E) and a positive integer k, an edge modification problem for a graph property Π consists in deciding whether there exists a set F of pairs of V of size at most k such that the graph \(H=(V,E\vartriangle F)\) satisfies the property Π. In the Π edge-completion problem, the set F is constrained to be disjoint from E; in the Π edge-deletion problem, F is a subset of E; no constraint is imposed on F in the Π edge-editing problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity (Cai in Inf. Process. Lett. 58:171–176, 1996; Fellows et al. in FCT, pp. 312–321, 2007; Heggernes et al. in STOC, pp. 374–381, 2007). When parameterized by the size k of the set F, it has been proved that if Π is a hereditary property characterized by a finite set of forbidden induced subgraphs, then the three Π edge-modification problems are FPT (Cai in Inf. Process. Lett. 58:171–176, 1996). It was then natural to ask (Bodlaender et al. in IWPEC, 2006) whether these problems also admit a polynomial kernel. in polynomial time to an equivalent instance (G′,k′) with size bounded by a polynomial in k). Using recent lower bound techniques, Kratsch and Wahlström answered this question negatively (Kratsch and Wahlström in IWPEC, pp. 264–275, 2009). However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. question to characterize for which type of graph properties, the parameterized edge-modification problems have polynomial kernels. Kratsch and Wahlström asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P 4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that Parameterized cograph edge-modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the P l -free edge-deletion and the C l -free edge-deletion problems for l⩾7 and l≥4 respectively. Indeed, if they exist, then NP⊆coNP/poly.

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References

  1. 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)

    Article  MATH  Google Scholar 

  2. Bessy, S., Paul, C., Perez, A.: Polynomial kernels for 3-leaf power graph modification problems. Discrete Appl. Math. 158(16), 1732–1744 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bodlaender, H.L., Cai, L., Chen, J., Fellows, M.R., Telle, J.A., Marx, D.: Open problems in parameterized and exact computation. In: International Workshop on Parameterized and Exact Computation (IWPEC) (2006)

    Chapter  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: Symposium on Theoretical Aspects of Computer Science (STACS). Leibniz International Proceedings in Informatics, vol. 9, pp. 165–176 (2011)

    Google Scholar 

  6. Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412(35), 4570–4578 (2011)

    Article  MATH  Google Scholar 

  7. Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications (1999)

    Book  MATH  Google Scholar 

  8. Brügmann, D., Komusiewicz, C., Moser, H.: On generating triangle-free graphs. Electron. Notes Discrete Math. 32, 51–58 (2009)

    Article  Google Scholar 

  9. Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)

    Article  MATH  Google Scholar 

  10. Chen, J., Meng, J.: A 2k kernel for the cluster editing problem. In: International Computing and Combinatorics Conference (COCOON). Lecture Notes in Computer Science, vol. 6196, pp. 459–468 (2010)

    Chapter  Google Scholar 

  11. Courcelle, B., Engelfriet, J., Rozenberg, G.: Handle-rewriting hypergraph grammars. J. Comput. Syst. Sci. 46(2), 218–270 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cunningham, W., Edmonds, J.: A combinatorial decomposition theory. Can. J. Math. 32(3), 734–765 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. Downey, R., Fellows, M.: Parameterized Complexity. Springer, Berlin (1999)

    Book  Google Scholar 

  14. El-Mallah, E.S., Colbourn, C.: The complexity of some edge deletion problems. IEEE Trans. Circuits Syst. 35(3), 354–362 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fellows, M., Langston, M., Rosamond, F., Shaw, P.: Efficient parameterized preprocessing for cluster editing. In: Fundamentals of Computation Theory, 16th International Symposium (FCT). Lecture Notes in Computer Science, vol. 4639, pp. 312–321 (2007)

    Chapter  Google Scholar 

  16. Flum, J., Grohe, M.: Parameterized Complexity Theory. Text in Theoretical Computer Science. Springer, Berlin (2006)

    Google Scholar 

  17. Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Garey, M., Johnson, S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1978)

    Google Scholar 

  19. Golumbic, M., Kaplan, H., Shamir, R.: Graph sandwich problems. J. Algorithms 19, 449–473 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Habib, M., Paul, C.: A survey on algorithmic aspects of modular decomposition. Comput. Sci. Rev. 4(1), 41–59 (2010)

    Article  Google Scholar 

  21. Heggernes, P., Paul, C., Telle, J.A., Villanger, Y.: Interval completion with few edges. In: ACM Symposium on Theory of Computing (STOC), pp. 374–381 (2007)

    Google Scholar 

  22. Kenyon-Mathieu, C., Schudy, W.: How to rank with few errors. In: ACM Symposium on Theory of Computing (STOC), pp. 95–103 (2007)

    Google Scholar 

  23. Kratsch, S., Wahlström, M.: Two edge modification problems without polynomial kernels. In: International Workshop on Parameterized and Exact Computation (IWPEC). Lecture Notes in Computer Science, vol. 5917, pp. 264–275 (2009)

    Chapter  Google Scholar 

  24. Liu, Y., Wang, J., Guo, J., Chen, J.: Cograph editing: Complexity and parameterized algorithms. In: International Computing and Combinatorics Conference (COCOON). Lecture Notes in Computer Science, vol. 6842, pp. 110–121 (2011)

    Chapter  Google Scholar 

  25. Ma, T.-H., Spinrad, J.: An O(n 2) algorithm for undirected split decomposition. J. Algorithms 16, 145–160 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Natanzon, A., Shamir, R., Sharan, R.: Complexity classification of some edge modification problems. Discrete Appl. Math. 113(1), 109–128 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford Lectures Series in Mathematics and Its Applications, vol. 31. Oxford University Press, London (2006)

    Book  MATH  Google Scholar 

  28. Niedermeier, R., Rossmanith, P.: A general method to speed up fixed-parameter-tractable algorithms. Inf. Process. Lett. 73(3–4), 125–129 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Oum, S.: Graphs of bounded rank-width. PhD thesis, Princeton University (2005)

  30. Rose, D.: A graph-theoretic study of the numerical solution of sparse positive systems of linear equations. In: Graph Theory and Computing, pp. 183–217 (1972)

    Google Scholar 

  31. Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144(1–2), 173–182 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Tarjan, R., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13, 566–579 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  33. Thomassé, S.: A 4k 2 kernel for feedback vertex set. ACM Trans. Algorithms 6(2) (2010)

  34. van Zuylen, A., Williamson, D.P.: Deterministic pivoting algorithms for constrained ranking and clustering problems. Math. Oper. Res. 34(3), 594–620 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for fruitful comments that improved the presentation of this paper. Research supported by the AGAPE project (ANR-09-BLAN-0159).

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Authors and Affiliations

  1. Department of Computer Science, Iowa State University, Ames, IA, 50011, USA

    Sylvain Guillemot

  2. INRIA, CNRS, Université Nice–Sophia-Antipolis, Sophia-Antipolis, France

    Frédéric Havet

  3. LIRMM, Université Montpellier II, CNRS, Montpellier, France

    Christophe Paul

  4. LIFO, Université d’Orléans, Orleans, France

    Anthony Perez

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  1. Sylvain Guillemot
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  2. Frédéric Havet
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Correspondence to Anthony Perez.

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Guillemot, S., Havet, F., Paul, C. et al. On the (Non-)Existence of Polynomial Kernels for P l -Free Edge Modification Problems. Algorithmica 65, 900–926 (2013). https://doi.org/10.1007/s00453-012-9619-5

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  • DOI: https://doi.org/10.1007/s00453-012-9619-5

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