Faster Fixed Parameter Tractable Algorithms for Undirected Feedback Vertex Set

  • Venkatesh Raman
  • Saket Saurabh
  • C. R. Subramanian
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2518)

Abstract

We give a O(maxû12k, (4lgk)ký · nω) algorithm for testing whether an undirected graph on n vertices has a feedback vertex set of size at most k where O(nω) is the complexity of the best matrix multiplication algorithm. The previous best fixed parameter tractable algorithm for the problem took O((2k + 1)k n2) time. The main technical lemma we prove and use to develop our algorithm is that that there exists a constant c such that, if an undirected graph on n vertices with minimum degree 3 has a feedback vertex set of size at most c√n, then the graph will have a cycle of length at most 12. This lemma may be of independent interest.

We also show that the feedback vertex set problem can be solved in O(dkkn) for some constant d in regular graphs, almost regular graphs and (fixed) bounded degree graphs.

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References

  1. 1.
    J. Alber, H. L. Bodlaender, H. Fernau and R. Niedermeier, ‘Fixed Parameter Algorithms for Dominating Set and Related Problems on Planar Graphs’, Lecture Notes in Computer Science 1851 (2000) 93–110; to appear in Algorithmica. MathSciNetGoogle Scholar
  2. 2.
    J. Alber, H. Fan, M. R. Fellows, H. Fernau, R. Niedermeier, F. Rosamand and U. Stege, ‘Refined Search Tree Techniques for Dominating Set on Planar Graphs’, Lecture Notes in Computer Science 2136 (2001) 111–122.CrossRefGoogle Scholar
  3. 3.
    N. Bansal and V. Raman, ‘Upper Bounds for MAX-SAT further improved, Lecture Notes in Computer Science 1741 (1999) 247–258.MathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Bar-Yehuda, D. Geiger, J. Naor, R. M. Roth, ‘Approximation Algorithms for the Feedback Vertex Set Problem with Applications to Constraint Satisfaction and Bayesian Inference’, Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, Arlington, Virginia, (1994) 344–354.Google Scholar
  5. 5.
    A Becker, R. Bar-Yehuda and D. Geiger, ‘Random Algorithms for the Loop Cutset Problem’, Journal of Artificial Intelligence Research 12 (2000) 219–234.MATHMathSciNetGoogle Scholar
  6. 6.
    J. Chen, I. A. Kanj and W. Jia, ‘Vertex Cover, Further Observations and Further Improvements’, Lecture Notes in Computer Science 1665 (1999) 313–324.MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Chen, D. K. Friesen, W. Jia and I. A. Kanj, ‘Using Nondeterminism to Design Efficient Deterministic Algorithms’ in Proceedings of 21st Foundations of Software Technology and Theoretical Computer Science (FST TCS) conference, Lecture Notes in Computer Science, Springer Verlag 2245 (2001) 120–131.CrossRefGoogle Scholar
  8. 8.
    R. Downey and M. R. Fellows, ‘Parameterized Complexity’, Springer Verlag 1998.Google Scholar
  9. 9.
    P. Erdos and L. Posa, ‘On the maximal number of disjoint circuits of a graph’, Publ Math. Debrecen 9 (1962) 3–12.MathSciNetGoogle Scholar
  10. 10.
    A. Itai and M. Rodeh, “Finding a minimum circuit in a graph”, SIAM Journal on Computing, 7 (1978) 413–423.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    R. Niedermeier and P. Rossmanith, ‘Upper Bounds for Vertex Cover: Further Improved’, in Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS), Lecture Notes in Computer Science 1563 (1999) 561–570.Google Scholar
  12. 12.
    R. Niedermeier and P. Rossmanith, ‘New Upper Bounds for Maximum Satisfiability’, Journal of Algorithms 36 (2000) 63–68.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    V. Raman, ‘Parameterized Complexity,’ in Proceedings of the 7th National Seminar on Theoretical Computer Science, Chennai, India (1997), 1–18.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Venkatesh Raman
    • 1
  • Saket Saurabh
    • 2
  • C. R. Subramanian
    • 1
  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.Chennai Mathematical InstituteChennaiIndia

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