Advertisement

Algorithmica

, Volume 52, Issue 2, pp 133–152 | Cite as

Fixed-Parameter Complexity of Minimum Profile Problems

  • Gregory Gutin
  • Stefan Szeider
  • Anders Yeo
Article

Abstract

The profile of a graph is an integer-valued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NP-hard problem, we consider parameterized versions of the problem. Namely, we study the problem of deciding whether the profile of a connected graph of order n is at most n−1+k, considering k as the parameter; this is a parameterization above guaranteed value, since n−1 is a tight lower bound for the profile. We present two fixed-parameter algorithms for this problem. The first algorithm is based on a forbidden subgraph characterization of interval graphs. The second algorithm is based on two simple kernelization rules which allow us to produce a kernel with linear number of vertices and edges. For showing the correctness of the second algorithm we need to establish structural properties of graphs with small profile which are of independent interest.

Keywords

Graph profile Fixed parameter tractability Above guaranteed value Kernel 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13, 335–379 (1976) zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38, 31–45 (2007) CrossRefGoogle Scholar
  3. 3.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999) Google Scholar
  4. 4.
    Billionnet, A.: On interval graphs and matrix profiles. RAIRO Tech. Oper. 20, 245–256 (1986) zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hallett, M.T., Wareham, H.T.: Parameterized complexity analysis in computational biology. Comput. Appl. Biosci. 11, 49–57 (1995) Google Scholar
  6. 6.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996) zbMATHCrossRefGoogle Scholar
  7. 7.
    Diaz, J., Gibbons, A., Paterson, M., Toran, J.: The minsumcut problem. Lect. Notes Comput. Sci. 519, 65–79 (1991) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  9. 9.
    Fomin, F.V., Golovach, P.A.: Graph searching and interval completion. SIAM J. Discrete Math. 13, 454–464 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.R.: Computers and Intractability. Freeman, New York (1979) zbMATHGoogle Scholar
  11. 11.
    Goldberg, P.W., Golumbic, M.C., Kaplan, H., Shamir, R.: Four strikes against physical mapping of DNA. J. Comput. Biol. 2, 139–152 (1995) Google Scholar
  12. 12.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38, 31–45 (2007) CrossRefGoogle Scholar
  13. 13.
    Gutin, G., Rafiey, A., Szeider, S., Yeo, A.: The linear arrangement problem parameterized above guaranteed value. Theory Comput. Syst. 41, 521–538 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Heggernes, P., Paul, C., Telle, J.A., Villanger, Y.: Interval completion with few edges. In: Proc. STOC 2007, 39th ACM Symposium on Theory of Computing, pp. 374–381. ACM, New York (2007) CrossRefGoogle Scholar
  15. 15.
    Karp, R.M.: Mapping the genome: some combinatorial problems arising in molecular biology. In: Proc. 25th Annual Symp. Theory Comput., pp. 278–285 (1993) Google Scholar
  16. 16.
    Kendall, D.G.: Incidence matrices, interval graphs, and seriation in archeology. Pac. J. Math. 28, 565–570 (1969) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Lekkerkerker, C.G., Boland, J.C.: Representation of a finite graph by a set of intervals on the real line. Fund. Math. Pol. Akad. Nauk 51, 45–64 (1962) zbMATHMathSciNetGoogle Scholar
  18. 18.
    Lin, Y., Yuan, J.: Profile minimization problem for matrices and graphs. Acta Math. Appl. Sin. 10, 107–112 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, London (2006) zbMATHGoogle Scholar
  20. 20.
    Serna, M., Thilikos, D.M.: Parameterized complexity for graph layout problems. EATCS Bull. 86, 41–65 (2005) MathSciNetzbMATHGoogle Scholar
  21. 21.
    West, D.B.: Introduction to Graph Theory. Prentice Hall, New York (2001) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Computer ScienceRoyal Holloway University of LondonEghamUK
  2. 2.Department of Computer ScienceUniversity of HaifaHaifaIsrael
  3. 3.Department of Computer ScienceDurham UniversityDurhamUK

Personalised recommendations