A Pseudo ∈-Approximate Algorithm For Feedback Vertex Set

  • Tianbing Qian
  • Yinyu Ye
  • Panos M. Pardalos
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 7)


While the picture of approximation complexity class becomes clear for most combinatorial optimization problems, it remains an open question whether Feedback Vertex Set can be approximated within a constant ratio in directed graph case. In this paper we present an approximation algorithm with performance bound L max −1, where L max is the largest length of essential cycles in the graph G(V,E). The worst case bound is \(\left\lfloor {\sqrt {{{\left| V \right|}^2} - \left| V \right| - \left| E \right| + 1} } \right\rfloor \) which, in general, is inferior to Seymour’s recent result [14], but becomes a small constant for some graphs. Furthermore, we prove the so-called pseudo ∈-approximate property, i.e. FVS can be divided into a class of disjoint NP -complete subproblems, and our heuristic becomes ∈-approximate for each one of these subproblems.


approximation bound feedback vertex set NP -complete 


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  1. 1.
    Bafna, V., Berman, P., and Fujito, T., Approximating Feedback Vertex Set for Undirected Graphs within Ratio 2, Manuscript, (1994).Google Scholar
  2. 2.
    Balas, E., A Sharp Bound On The Relation Between Optimal Integer And Fractional Covers, Mathematics Of Operations Research, 9, pp. 1–7, (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Chakradhar, S., Balakrishnan, A., Agrawal, V., An Exact Algorithm For Selecting Partial Scan Flip-Flops, Manuscript, (1994).Google Scholar
  4. 4.
    Chvatal, V., A Greedy Heuristic For The Set Covering, Mathematics Of Operations Research, 1, pp. 515–531, (1979).Google Scholar
  5. 5.
    Garey, M. R. and Johnson, D. S., Computers And Reducibility -A Guide To The Theory Of NP-Completeness, W. H. Freeman, San Francisco, (1979).Google Scholar
  6. 6.
    Hochbaum, D., Approximation Algorithms For Set Covering And Vertex Cover Problem, SIAM Journal on Computing, 11, pp. 555–556, (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Johnson, D.B., Finding all the elementary circuits of a directed graph, SIAM J. Computing, Vol. 4, No. 1, pp. 77–84, (1975).zbMATHCrossRefGoogle Scholar
  8. 8.
    Johnson, D.S., Approximation Algorithms For Combinatorial Problems, Journal Of Computer And System Science, 9, pp. 256–298, (1974).zbMATHCrossRefGoogle Scholar
  9. 9.
    Karp, R. M., Reducibility Among Combinatorial Problems, Complexity Of Computer Computations, R.E. Miller and J.W. Thatcher eds, Plenum Press, pp. 85–103, (1972).Google Scholar
  10. 10.
    Leighton, T. and Rao, S., An Approximate Max-Flow Min-Cut Theorem for Uniform Multicommodity Flow Problems with Applications to Approximation Algorithms, Manuscript, (1993).Google Scholar
  11. 11.
    Levy, H. and Lowe, L., A Contraction Algorithm For Finding Small Cycle Cutsets, Journal Of Algorithm, 9, pp. 470–493, (1988).zbMATHCrossRefGoogle Scholar
  12. 12.
    Lund, C. and Yannakakis, M., On The Hardness Of Approximating Minimization Problems, Proceedings Of the 25th ACM Symp. On Theory Of Computing, pp. 286–293, (1993).Google Scholar
  13. 13.
    Papadimitriou, C. and Yannakakis, M., Optimization, approximation and complexity classes, Proc. of the 20th Annual ACM Symp. on Theory of Computing, pp. 251–277, (1988).Google Scholar
  14. 14.
    Seymour, P.D., Packing Directed Circuits Fractionally, to appear in Combinatorica, (1993).Google Scholar
  15. 15.
    Shamir, A., A Linear Time Algorithm For Finding Minimum Cutsets In Reduced Graphs, SIAM Journal On Computing, Vol. 8, No. 4, pp. 645–655, (1979).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Tarjan, R.E., Depth First Search And Linear Graph Algorithms, SIAM Journal on Computing, 1, pp. 146–160, (1972).MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Wang, C., Lloyd, E. and Soffa, M., Feedback Vertex Sets And Cyclically Reducible Graphs, Journal Of The Association For Computing Machinery, Vol. 32, No. 2, pp. 296–313, (1985).MathSciNetzbMATHGoogle Scholar
  18. 18.
    Yannakakis, M., Some Open Problems in Approximation, Proc. of the second Italian Conference on Algorithm and Complexity, CIAC’94, pp. 33–39, Italy, Feb. (1994).Google Scholar
  19. 19.
    Yehuda, B. and Even, S., A Linear Time Approximation Algorithm For The Weighted Vertex Cover Problem, Journal Of Algorithms, 2, pp. 198–203, (1981).MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Yehuda, B., Geiger, D., Naor, J., and Roth, R.M., Approximation Algorithms for the Vertex Feedback Set Problem with Applications to Constraint satisfaction and bayesian inference, Proc. of the 5th Annual ACm-SIAM Symp. on Discrete Algorithms, pp. 344–354, (1994).Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Tianbing Qian
    • 1
  • Yinyu Ye
    • 1
  • Panos M. Pardalos
    • 2
  1. 1.Department of Management ScienceThe University of IowaIowa CityUSA
  2. 2.Center for Applied Optimization and Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

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