Algorithmica

, Volume 52, Issue 2, pp 203–225 | Cite as

Short Cycles Make W-hard Problems Hard: FPT Algorithms for W-hard Problems in Graphs with no Short Cycles

Article

Abstract

We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles.

More specifically, we give fixed parameter tractable algorithms for Dominating Set, t-Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs with girth at least five. These problems are known to be W[i]-hard for some i≥1 in general graphs. We also show that the Dominating Set problem is W[2]-hard for bipartite graphs and hence for triangle free graphs.

In the case of Independent Set and several of its variants, we show these problems to be fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem where one is interested in finding an induced subgraph on k vertices having at least l edges, parameterized by k, is W[1]-hard even on graphs with girth at least six.

Finally, we give an O(log p) ratio approximation algorithm for the Dominating Set problem for graphs with girth at least 5, where p is the size of an optimum dominating set of the graph. This improves the previous O(log n) factor approximation algorithm for the problem, where n is the number of vertices of the input graph.

Keywords

Dominating set Independent set Set cover t-vertex cover Parameterized complexity 

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References

  1. 1.
    Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial time data reduction for dominating set. J. ACM 51(3), 363–384 (2004) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Alber, J., Fan, H., Fellows, M.R., Fernau, H., Niedermeier, R., Rosamond, F., Stege, U.: A refined search tree technique for dominating set on planar graphs. J. Comput. Syst. Sci. 71, 385–405 (2005) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alekseev, V.E.: On easy and hard hereditary classes of graphs with respect to the independent set problem. Discrete Appl. Math. 132(1–3), 17–26 (2003) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alekseev, V.E., Korobitsyn, D.V., Lozin, V.V.: Boundary classes of graphs for the dominating set problem. Discrete Math. 285(1–3), 1–6 (2004) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bläser, M.: Computing small partial coverings. Inf. Process. Lett. 85(6), 327–331 (2003) CrossRefMATHGoogle Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Threshold dominating sets and an improved characterization of W[2]. Theor. Comput. Sci. 209(1–2), 123–140 (1998) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999) Google Scholar
  8. 8.
    Downey, R.G., Fellows, M.R., Vardy, A., Whittle, G.: The parametrized complexity of some fundamental problems in coding theory. SIAM J. Comput. 29(2), 545–570 (1999) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R., Raman, V.: The complexity of irredundant sets parameterized by size. Discrete Appl. Math. 100(3), 155–167 (2000) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Duh, R., Fürer, M.: Approximation of k-set cover by semi-local optimization. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC), pp. 256–264 (1997) Google Scholar
  11. 11.
    Edmonds, J.: Paths, trees and flowers. Can. J. Math. 17, 449–467 (1965) MATHMathSciNetGoogle Scholar
  12. 12.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fernau, H.: Parameterized algorithms: a graph-theoretic approach. Habilitationsschrift, Universität Tübingen, Tübingen, Germany (2005) Google Scholar
  14. 14.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  15. 15.
    Fomin, F.V., Gaspers, S., Pyatkin, A.V.: Finding a minimum feedback vertex set in time O(1.7548n). In: Proceedings of the 2nd International Workshop on Parameterized and Exact Computation (IWPEC). Lecture Notes in Computer Science, vol. 4169, pp. 184–191. Springer, Berlin (2006) CrossRefGoogle Scholar
  16. 16.
    Guo, J., Niedermeier, R., Wernicke, S.: Parameterized complexity of generalized vertex cover problems. In: Proceeding of the 9th International Workshop Algorithms and Data Structures (WADS). Lecture Notes in Computer Science, vol. 3608, pp. 36–48. Springer, Berlin (2005) Google Scholar
  17. 17.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9(3), 256–278 (1974) MATHCrossRefGoogle Scholar
  18. 18.
    Jukna, S.: Extremal Combinatorics. Springer, Berlin (2001) MATHGoogle Scholar
  19. 19.
    Khot, S., Raman, V.: Parameterized complexity of finding subgraphs with hereditary properties. Theor. Comput. Sci. 289(2), 997–1008 (2002) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kortsarz, G., Peleg, D.: On choosing a dense subgraph. In: Proceeding of the 34th Annual Symposium on Foundations of Computer Science (FOCS), pp. 692–701 (1993) Google Scholar
  21. 21.
    Lovàsz, L.: On the ratio of optimal fractional and integral covers. Discrete Math. 13, 383–390 (1975) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, London (2006) MATHGoogle Scholar
  23. 23.
    Raman, V., Saurabh, S.: Triangles, 4-cycles and parameterized (in-)tractability. In: Proceeding of the 10th Scandinavian Workshop on Algorithm Theory (SWAT). Lecture Notes in Computer Science, vol. 4059, pp. 304–315. Springer, Berlin (2006) Google Scholar
  24. 24.
    Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.The Institute of Mathematical SciencesChennaiIndia

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