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Feedback Set Problems

  • Paola Festa
  • Panos M. Pardalos
  • Mauricio G. C. Resende

Abstract

Not long ago, there appeared to be a consensus in the literature that feedback set problems, which originated from the area of combinational circuit design, were the least understood among all the classical combinatorial optimization problems due to the lack of positive results in efficient exact and approximating algorithms. This picture has been totally changed in recent years. Dramatic progress has occurred in developing approximation algorithms with provable performance; new bounds have been established one after the other and it is probably fair to say that feedback set problems are becoming among the most exciting frontend problems in combinatorial optimization.

Keywords

Approximation Algorithm Greedy Randomize Adaptive Search Procedure Interval Graph Permutation Graph Deadlock Prevention 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    V. Bafna, P. Berman, and T. lijito, Constant ratio approximations of the weighted feedback vertex set problem for undirected graphs, in ISAAC95, Algorithms and Computation,J. Staples, P. Eades, N. Katoh and A. Moffat Eds., Lecture Notes in Computer Science Vol.1004, Springer-Verlag (1995) pp. 142–151.Google Scholar
  2. [2]
    R. Bar-Yehuda, D. Geiger, J. Naor, and R.M. Roth, Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and Bayesian inference, A preliminary version of this paper appeared in the Proc. of the 5t h Annual ACM-SIAM Symp. on Discrete Algorithms, pp. 344–354, (1994) and subsequently published in SIAM J. Comput. Vol. 27 No. 4 (1998) pp. 942–959.Google Scholar
  3. [3]
    A. Becker, and D. Geiger, Approximation algorithm for the loop cut-set problem, in Proceedings of the 10th Conference on Uncertainty in Artificial Intelligence Morgan Kaufman (1994) pp. 60–68.Google Scholar
  4. [4]
    A. Becker, and D. Geiger, Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem, Artificial Intelligence Vol. 83 (1996) pp. 167–188.MathSciNetCrossRefGoogle Scholar
  5. [5]
    J. A. Bondy, G. Hopkins, and W. Staton, Lower bounds for induced forests in cubic graphs, Gonad. Math. Bull. Vol. 30 (1987) pp. 193–199.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    D.P. Bovet, S. de Agostino, and R. Petreschi, Parallelism and the feedback vertex set problem, Information Processing Letters Vol. 28 (1988) pp. 81–85.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    A. Brandstädt, and D. Kratsch, On the restriction of some NP-complete graph problems to permutation graphs, in Proc. of Fundamentals of Computing Theory, Lecture Notes in Comp. Sci. Vol.199 ( L. Budach, Ed. Springer-Verlag, Berlin, 1985 ) pp. 53–62.CrossRefGoogle Scholar
  8. [8]
    A. Brandstädt, On improved time bounds for permutation graph problems, in Proc. of the 18 th Workshop on Graph-theoretic concepts in computer science, (Wiesbaden-Naurod, 1992) Springer-Verlag LNCS 657 (1993) pp. 1–10.Google Scholar
  9. [9]
    M.A. Breuer, and R. Gupta, BALLAST: A methodology for partial scan design, in Proc. of the 19 th Int. Symposium on Fault-Tolerant Computing (1989) pp. 118–125 Google Scholar
  10. [10]
    M. Cai, X. Deng, and W. Zang, A TDI system and its application to approximation algorithm, in Proc. of the 39th Annual Symposium on Foundations of Computer Science, Palo Alto, California, November 8–11, (1998).Google Scholar
  11. [11]
    M. Cai, X. Deng, and W. Zang, A min-max theorem on feedback vertex sets, to appear in Integer Programming and Combinatorial Optimization: Proceedings 7th International IPCO Conference, Lecture Notes in Computer Science Springer-Verlag (1999) Google Scholar
  12. [12]
    S Chakradhar, A. Balakrishnan, and V. Agrawal, An exact algorithm for selecting partial scan flip-flops, Manuscript (1994).Google Scholar
  13. [13]
    M.S. Chang, Y.D. Liang, Minimum feedback vertex sets in cocomparability graphs and convex bipartite graphs, Acta Informatica Vol. 34 (1997) pp. 337–346.MathSciNetCrossRefGoogle Scholar
  14. [14]
    I. Charon, A. Guenoche, O. Hudry, and F. Wairgard, New results on the computation of median orders, Discr. Math. Vol. 165 /166 (1997) pp. 139–153.CrossRefGoogle Scholar
  15. [15]
    R. Chen, X. Guo, and F. Zhang, The z-transformation graphs of perfect matchings of hexagonal system, Discr. Math. Vol. 72 (1988) pp. 405–415.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    K.T. Cheng, and V.D. Agrawal, A partial scan method for sequential circuits with feedback, IEEE Transactions on Computers Vol. 39 No. 4 (1990) pp. 544–548.CrossRefGoogle Scholar
  17. [17]
    Chin Lung Lu, and Chuan Yi Tang, A linear-time algorithm for the weighted feedback vertex problem on interval graphs, Information Processing Letters Vol. 61 (1997) pp. 107–111.MathSciNetCrossRefGoogle Scholar
  18. [18]
    F.A. Chudak, M.X. Goemans, D. Hochbaum, and D.P. Williamson, A primal-dual interpretation of two 2-approximation algorithms for the feedback vertex set problem in undirected graphs, Operations Research Letters Vol.22 (1998) pp.111–118. Vol. 4 (1979) pp. 233–235.Google Scholar
  19. [19]
    V. Chvatal, A greedy heuristic for the set covering problem, Mathematics Of Operations Research Vol. 4 (1979) pp. 233–235.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    S.R. Coorg, and C.P. Rangan, Feedback vertex set on cocomparability graphs, Networks Vol. 26 (1995) pp. 101–111.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    D.G. Corneil, and J. Fonlupt, The complexity of generalized clique covering, Discr. Appl. Math. Vol. 22 (1988) pp. 109–118.MathSciNetCrossRefGoogle Scholar
  22. [22]
    R. Dechter, and J. Pearl, The cycle cutset method for improving search performance in AI, in Proc. of the 3 th IEEE on AI Applications Orlando FL (1987).Google Scholar
  23. [23]
    R. Dechter, Enhancement schemes for constraint processing: Back-jumping, learning, and cutset decomposition, Artif. Intell. Vol. 41 (1990) pp. 273–312.MathSciNetCrossRefGoogle Scholar
  24. [24]
    J. Donald, J. Elwin, R. Hager, and P. Salamon, A bad example for the minimum feedback vertex set problem, IEEE Transactions on Circuits and Systems Vol. 32 (1995) pp. 491–493.CrossRefGoogle Scholar
  25. [25]
    R.G. Downey, and M.R. Fellows, Fixed-parameter tractability and completeness I: Basic results, SIAM Journal on Computing Vol. 24 (1995) pp. 873–921.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    P. Erdös, and L. Posa, On the maximal number of disjoint circiuts of a graph, Pubbl. Math. Debrecen Vol. 9 (1962) pp. 3–12.MATHGoogle Scholar
  27. [27]
    G. Even, S. Naor, B. Schieber, and L. Zosin, Approximating minimum subset feedback sets in undirected graphs, with applications, in the Proc. of the 4th Israel Symposium on Theory of Computing and Systems (1996) pp. 78–88.Google Scholar
  28. [28]
    G. Even, S. Naor, B. Schieber, and M. Sudan, Approximating minimum feedback sets and multicuts in directed graphs, Algorithmica Vol. 20 (1998) pp. 151–174.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    G. Even, J.S. Naor, and L. Zosin An 8-Approximation Algorithm for the Subset Feedback Vertex Problem, 37th Symp. on Foundations of Comp. Sci. ( FOCS ) (1996) pp. 310–319.Google Scholar
  30. [30]
    P. Festa, P.M. Pardalos, and M.G.C. Resende, Fortran subroutines for approximate solution of feedback set problems using GRASP, Manuscript, AT & -T Labs Research, Florham Park, NJ (1999).Google Scholar
  31. [31]
    M. Funke, and G. Reinelt, A polyhedral approach to the feedback vertex set problem, Manuscript (1996).Google Scholar
  32. [32]
    M.R. Carey, and D.S. Johnson, Computers And Intractability —A Guide to the Theory of NP-Completeness, W. H. Freeman, San Francisco, (1979).Google Scholar
  33. [33]
    M.R. Garey, and R.E. Tarjan, A linear-time algorithm for finding all feedback vertices, Information Processing Letters Vol.7 (1978) pp. 274276.Google Scholar
  34. [34]
    N. Garg, V.V. Vazirani, and M. Yannakakis, Approximate max-flow min-(multi) cut theorems and their applications, SIAM Journal on Computing Vol. 25 No. 2 (1996) pp. 235–251.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    F. Gavril, Some NP-complete problems on graphs, in Proceedings of the 11 th Conference on Information Science and Systems, John Hopkins Univ. Baltimore, Md., (1977) pp. 91–95.Google Scholar
  36. [36]
    M.X. Goemans, and D.P. Williamson, Primal-dual approximation algorithms for feedback problems in planar graphs, in Proceedings of the 5 th MPS Conference on Integer Programming and Combinatorial Optimization (IPCO) (1996) pp. 147–161.Google Scholar
  37. [37]
    M. Grötschel, L. Lovâsz, and A. Schrijver, Geometric algorithms and combinatorial optimization, Springer-Verlag, Berlin (1988) pp. 253–254.MATHCrossRefGoogle Scholar
  38. [38]
    M. Grötschel, and L. Lovâsz, Combinatorial optimization: A survey, DIMACS Technical Report 93–29 DIMACS Rutgers University (1993).Google Scholar
  39. [39]
    F. Harary, D.J. Klein, and T.P. Zivkovic, Graphical properties of poly-hexes: Perfect matching vector and forcing, J. Math. Chem. Vol. 6 (1991) pp. 295–306.MathSciNetCrossRefGoogle Scholar
  40. [40]
    D. Hochbaum, Approximation algorithms for set covering and vertex cover problem, SIAM Journal on Computing Vol.11 No. 3 (1982) pp. 555–556.MathSciNetCrossRefGoogle Scholar
  41. [41]
    T.C. Hu, Multi-commodity network flows, Operations Research Vol.11 (1963) pp. 344–360.Google Scholar
  42. [42]
    G. Isaak, Tournaments as feedback arc sets, Electronic Journal of Combinatorics Vol. 20 No. 2 (1995) pp. 1–19.Google Scholar
  43. [43]
    D.B. Johnson, Finding all the elementary circuits of a directed graph, SIAM J. Computing, Vol. 4, No. 1 (1975) pp. 77–84.MATHCrossRefGoogle Scholar
  44. [44]
    D.S. Johnson, Approximation algorithms for combinatorial problems, Journal of Computer and System Science Vol. 9 (1974) pp. 256–278.MATHCrossRefGoogle Scholar
  45. [45]
    R.M. Karp, Reducibility among combinatorial problems, Complexity Of Computer Computations R.E. Miller and J.W. Thatcher, Eds., New York: Plenum Press (1972) pp. 85–103.Google Scholar
  46. [46]
    A.K. Kevorkian, General topological results on the construction of a minimum essential set of a directed graph, IEEE Trans. Circuits and Systems Vol. 27 (1980) pp. 293–304.MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    D.J. Klein, and M. Randié, Innate degree of freedom of a graph, J. Computat. Chem. Vol. 8 (1987) pp. 516–521.CrossRefGoogle Scholar
  48. [48]
    D.J. Klein, T.P. Zivkovié, and R. Valenti, Topological long-range order for resonating-valance-bond structures, Phys. Rev. B Vol. 43A (1991) pp. 723–727.CrossRefGoogle Scholar
  49. [49]
    A. Kunzmann, and H.J. Wunderlich, An analytical approach to the partial scan problem, J. of Electronic Testing: Theory and Applications Vold (1990) pp. 163–174.Google Scholar
  50. [50]
    H. Kim, and J. Perl, A computational model for combined causal and diagnostic reasoning in inference systems, in Proc. of the 8 th IJCAI, Morgan-Kaufmann, San Mateo, CA, (1983) pp. 190–193.Google Scholar
  51. [51]
    S.L. Lauritzen, and D.J. Spiegelhalter, Local computations with probabilities on graphical structures and their application to expert systems (with discussion), J. Roy. Stat. Soc. Ser.B Vol. 50 (1988) pp. 157–224.MathSciNetMATHGoogle Scholar
  52. [52]
    D. Lee, and S. Reedy, On determining scan flip-flops in partial scan designs, in Proceedings Of International Conference on Computer Aided Design (1990) pp. 322–325.Google Scholar
  53. [53]
    T. Leighton, and S. Rao, An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms, in Proceedings of the 29 th Annual Symposium on Fundations of Computer Science, (1988) pp. 422–431.Google Scholar
  54. [54]
    A. Lempel, and I. Cederbaum, Minimum feedback arc and vertex sets of a directed graph, IEEE Transactions on circuit theory CT-13 (1966) pp. 399–403.Google Scholar
  55. [55]
    H. Levy, and L. Lowe, A contraction algorithm for finding small cycle cutsets, Journal Of Algorithm Vol. 9 (1988) pp. 470–493.MATHCrossRefGoogle Scholar
  56. [56]
    X.Li, and F. Zhang, Hexagonal systems with forcing edges, Discr. Math. Vol.140 (1995) pp. 253–263 Google Scholar
  57. [57]
    Y.D. Liang, On the feedback vertex set problem in permutation graphs, Information Processing Letters, Vol. 52 (1994) pp. 123–129.MathSciNetMATHCrossRefGoogle Scholar
  58. [58]
    J. Liu, and C. Zhao, A new bound on the feedback vertex sets in cubic graphs, Discrete Mathematics Vol. 148 (1996) pp. 119–131.MathSciNetMATHCrossRefGoogle Scholar
  59. [59]
    E.L. Lloyd, M.L. Soffa, and C.C. Wang, On locating minimum feedback vertex sets, Journal of Computer and System Sciences Vol. 37 (1988) pp. 292–311.MathSciNetMATHCrossRefGoogle Scholar
  60. [60]
    C.L. Lucchesi, and D.H. Younger, A minimax theorem for directed graphs, J. London Math. Soc. Vol. 17 (1978) pp. 369–374.MathSciNetMATHCrossRefGoogle Scholar
  61. [61]
    F.L. Luccio, Almost exact minimum feedback vertex set in meshes and butterflies, Information Processing Letters Vol. 66 ( 1998, pp. 59–64.MathSciNetMATHCrossRefGoogle Scholar
  62. [62]
    C Lund, and M. Yannakakis, On the hardness of approximating minimization problems, Proceedings Of the 25th ACM Symp. On Theory Of Computing (1993) pp. 286–293.Google Scholar
  63. [63]
    M.V. Marathe, C.Pandu Rangan, and R. Ravi, Efficient algorithms for generalized clique covering on interval graphs, Discr. Appl. Math. Vol. 39 (1992) pp. 87–93.MATHCrossRefGoogle Scholar
  64. [64]
    B. Monien, and R. Schultz, Four approximation algorithms for the feedback vertex set problems, in Proc. of the 7 th Conference on Graph Theoretic Concepts of Computer Science, Hanser-Verlag, Munich (1981) pp. 315–326.Google Scholar
  65. [65]
    T. Orenstein, Z. Kohavi, and I. Pomeranz, An optimal algorithm for cycle breaking in directed graphs, J. of Electronic Testing: Theory and Applications Vol. 7 (1995) pp. 71–81.CrossRefGoogle Scholar
  66. [66]
    L. Pachter, and P. Kim, Forcing matchings on square grids, Discr. Math Vol. 190 (1998) pp. 287–294.MathSciNetMATHCrossRefGoogle Scholar
  67. [67]
    C. Papadimitriou, and M. Yannakakis, Optimization, approximation and complexity classes, in Proc. of the 20th Annual ACM Symp. on Theory of Computing (1988) pp. 251–277.Google Scholar
  68. [68]
    P.M. Pardalos, T. Qian, and M.G.C. Resende, A greedy randomized adaptive search procedure for feedback vertex set, J. Comb. Opt. Vol. 2 (1999) pp. 399–412.MathSciNetMATHCrossRefGoogle Scholar
  69. [69]
    D. Peleg, Local majority voting, small coalitions, and controlling monopolies in graphs: A review, in Proc. of the 3 th Colloquium on Structural Information and Communication Complexity (1996) pp. 152–169.Google Scholar
  70. [70]
    D. Peleg, Size bounds for dynamic monopolies, in Proc. of the 4th Colloquium on Structural Information and Communication Complexity (1997) Carleton Univ. Press, Ottawa, pp. 165–175.Google Scholar
  71. [71]
    J. Perl, Fusion, propagation and structuring in belief networks, Artif. Intell. Vol. 29 (1986) pp. 241–288.CrossRefGoogle Scholar
  72. [72]
    T. Qian, Y. Ye, and P.M. Pardalos, A Pseudo-e approximation algorithm for feedback vertex set, Recent Advances in Global Optimization, Floudas, C.A. and Pardalos, P.M., Eds., Kluwer Academic Publishing (1995) pp. 341–351.Google Scholar
  73. [73]
    V. Ramachandran, Finding a minimum feedback arc set in reducible flow graphs, Journal of Algorithms Vol. 9 (1988) pp. 299–313.MathSciNetMATHCrossRefGoogle Scholar
  74. [74]
    B. Rosen, Robust linear algorithms for cutsets, Journal of Algorithms Vol. 3 (1982) pp. 205–217.MathSciNetMATHCrossRefGoogle Scholar
  75. [75]
    P.D. Seymour, Packing directed circuits fractionally, Combinatorica Vol. 15 (1995) pp. 281–288.MathSciNetMATHCrossRefGoogle Scholar
  76. [76]
    A. Shamir, A linear time algorithm for finding minimum cutsets in reduced graphs, SIAM Journal On Computing Vol.8 No.4 (1979) pp. 645–655.Google Scholar
  77. [77]
    R.D. Shatcher, S.K. Andersen, and P. Szolovits, Global conditioning for probabilistic inference in belief networks, in Proc of the 10 th Conferences on Uncertainty in AI,Seattle, WA (1994) pp. 514-522.Google Scholar
  78. [78]
    A.C. Shaw, The logical design of operating systems, Prentice-Hall, Englewood Cliffs, NJ, (1974).MATHGoogle Scholar
  79. [79]
    D.A. Simovici, and G. Grigoras, Even initial feedback vertex set problem is NP-complete, Information Processing Letters Vol. 8 (1979) pp. 64–66.MathSciNetMATHCrossRefGoogle Scholar
  80. [80]
    G.W. Smith, and R.B. Walford, The identification of a minimal feedback vertex set of a directed graph, IEEE Transactions on Circuits and Systems Vol.CAS-22 No. 1, (1975) pp. 9–14.MathSciNetCrossRefGoogle Scholar
  81. [81]
    E. Speckenmeyer, On feedback vertex sets and nonseparating independent sets in cubic graphs, Journal of Graph Theory Vol. 12 (1988) pp. 405–412.MathSciNetMATHCrossRefGoogle Scholar
  82. [82]
    E. Speckenmeyer, On feedback problems in digraphs, in Lecture Notes in Computer Science, Springer-Verlag Vol. 411 (1989) pp. 218–231.MathSciNetGoogle Scholar
  83. [83]
    H. Stamm, On feedback problems in a planar digraph, in R. Möhring ed. Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, Springer-Verlag (1990) Vol. 484 pp. 79–89.Google Scholar
  84. [84]
    R.E. Tarjan, Depth first search and linear graph algorithms, SIAM Journal on Computing Vol.1 (1972) pp. 146–160.Google Scholar
  85. [85]
    S. Ueno, Y. Kajitani, and S. Gotoh, On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three, Discrete Mathematics Vol. 72 (1988) pp. 355–360.MathSciNetMATHCrossRefGoogle Scholar
  86. [86]
    V. Vazirani, Approximation Algorithms, Manuscript, College of Computing, Georgia Institute of Technology.Google Scholar
  87. [87]
    C. Wang, E. Lloyd, and M. Soffa, Feedback vertex sets and cyclically reducible graphs, Journal of the Association for Computing Machinery Vol. 32 No. 2 (1985) pp. 296–313.MathSciNetMATHCrossRefGoogle Scholar
  88. [88]
    M. Yannakakis, Node and edge-deletion NP-complete problems, in Proceedings of the 10t h Annual ACM Symposium on Theory of Computing (1978) pp. 253–264.Google Scholar
  89. [89]
    M. Yannakakis, and F. Gavril, The maximum k-colorable subgraph problem for chordal graphs, Info. Process. Lett Vol.24 (1987) pp. 133137.Google Scholar
  90. [90]
    M. Yannakakis, Some open problems in approximation, in Proc. of the second Italian Conference on Algorithm and Complexity, CIAC’94 Italy, Feb. (1994) pp. 33–39.Google Scholar
  91. [91]
    D.H. Younger, Minimum feedback arc set for a directed graph, IEEE Transactions on Circuit Theory Vol. CT-10 (1963) pp. 238–245.Google Scholar
  92. [92]
    M. Zheng, and X. Lu, On the maximum induced forests of a connected cubic graph without triangles, Discr. Math. Vol. 85 (1990) pp. 89–96.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Paola Festa
    • 1
  • Panos M. Pardalos
    • 2
  • Mauricio G. C. Resende
    • 3
  1. 1.Mathematics and Computer Science DepartmentUniversity of SalernoBaronissi (SA)Italy
  2. 2.Center for Applied Optimization Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA
  3. 3.Information Sciences Research Center AT & T Labs ResearchShannon LaboratoryFlorham ParkUSA

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