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Feedback Arc Set

Part of the book series: SpringerBriefs in Computer Science ((BRIEFSCOMPUTER))

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Abstract

The chapter presents what would in a nutshell be a traveling through time with the problem of Feedback Arc Set. The review goes from the first paper, jumping through a number of periods and research results, ending with recent achievements, including a quantum computing algorithm. This chapter, more than any other part of the book, provides a historical compendium of techniques in Computer Science.

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Notes

  1. 1.

    On 26 August 2021, under a name 815.gz, a file for the subroutines was still available for downloaded at: https://calgo.acm.org/.

  2. 2.

    The software was at the time of publication of the paper in question available at: http://www.enst.fr/~charon/tournament/median.html. It seems, however, that it is not available anymore, access was attempted on 27th of September 2021.

  3. 3.

    https://sdopt-tearing.readthedocs.io/en/latest/

  4. 4.

    Algorithms are implemented in Java, and on the 30th of January 2022 were available at: https://github.com/stamps/FAS.

  5. 5.

    Web page of the Laboratory of Web Algorithmics: http://law.di.unimi.it/datasets.php.

  6. 6.

    Dataset web-page accessed on 31th of January 2022 at: https://law.di.unimi.it/webdata/clueweb12/.

  7. 7.

    n-dimensional counterpart of a cube, or a square: https://mathworld.wolfram.com/HypercubeGraph.html.

  8. 8.

    Finding a specific path alongside edges of a hypercube [81].

  9. 9.

    A quantum unstructured search algorithm that finds desired item in \( O(\sqrt {N}) \) steps, with N being number of items to choose from [62].

  10. 10.

    C+ + code, including benchmark datasets, was on 4th of March 2022 available at: https://git.mpi-cbg.de/mosaic/software/math/FaspHeuristic.

References

  1. Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ailon, N.: Aggregation of partial rankings, p-ratings and top-m lists. Algorithmica 57(2), 284–300 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: ranking and clustering. J. ACM 55(5), 1–27 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alon, N., Lokshtanov, D., Saurabh, S.: Fast FAST. In: ICALP: International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 5555, pp. 49–58. Springer, Berlin, Heidelberg (2009)

    Google Scholar 

  5. Alon, N., Spencer, J.: The Probabilistic Method. Wiley, New York (1992)

    MATH  Google Scholar 

  6. Ambainis, A., Balodis, K., Iraids, J., Kokainis, M., Prusis, K., Vihrovs, J.: Quantum speedups for exponential-time dynamic programming algorithms. In: SODA ’19: Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1783–1793. Society for Industrial and Applied Mathematics (2019)

    Google Scholar 

  7. Arditti, D.: A new algorithm for searching for an order induced by pairwise comparisons. In: E.D. et al. (ed.) Data Analysis and Informatics III, pp. 323–343. North Holland, Amsterdam (1984)

    Google Scholar 

  8. Ariyoshi, H., Higashiyama, Y.: A heuristic algorithm for the minimum feedback arc set problem. Res. Inst. Math. Anal. 427, 112–130 (1981). Kyoto University Research Information Repository (Departmental Bulletin Paper)

    Google Scholar 

  9. Arora, S., Frieze, A., Kaplan, H.: A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. Math. Program. 92(1), 1–36 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Baharev, A., Schichl, H., Neumaier, A., Achterberg, T.: An exact method for the minimum feedback arc set problem. ACM J. Exp. Algorithm. 26, 1–28 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications. Springer. London (2002)

    Book  MATH  Google Scholar 

  12. Bang-Jensen, J., Maddaloni, A., Saurabh, S.: Algorithms and kernels for feedback set problems in generalizations of tournaments. Algorithmica 76(2), 320–343 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bar-Yehuda, R.: One for the price of two: a unified approach for approximating covering problems. Algorithmica 27(2), 131–144 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Barthelemy, J., Guenoche, A., Hudry, O.: Median linear orders: heuristics and a branch and bound algorithm. Eur. J. Oper. Res. 42(3), 313–325 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bartholdi, J., Tovey, C.A., Trick, M.A.: Voting schemes for which it can be difficult to tell who won the election. Soc. Choice Welfare 6(2), 157–165 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Berger, B., Shor, P.W.: Approximation algorithms for the maximum acyclic subgraph problem. In: SODA ’90: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 236–243. Society for Industrial and Applied Mathematics (1990)

    Google Scholar 

  17. Bessy, S., Fomin, F.V., Gaspers, S., Paul, C., Perez, A., Saurabh, S., Thomassé, S.: Kernels for feedback arc set in tournaments. J. Comput. Syst. Sci. 77(6), 1071–1078 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bhat, K.V., Kinariwala, B.: Optimum tearing in large scale systems and minimum feedback cutsets of a digraph. J. Franklin Inst. 307(2), 83–94 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Biegler, L.T., Grossmann, I.E., Westerberg, A.W.: Systematic Methods for Chemical Process Design. Prentice Hall PTR (1997)

    Google Scholar 

  20. Bodlaender, H.L., Fomin, F.V., Koster, A.M.C.A., Kratsch, D., Thilikos, D.M.: A note on exact algorithms for vertex ordering problems on graphs. Theory Comput. Syst. 50(3), 420–432 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Brandenburg, F.J., Hanauer, K.: Sorting heuristics for the feedback arc set problem—technical report mip-1104. Tech. rep., Department of Informatics and Mathematics, University of Passau, Germany (2011)

    Google Scholar 

  22. Brglez, F., Bryan, D., Kozminski, K.: Combinational profiles of sequential benchmark circuits. In: IEEE International Symposium on Circuits and Systems, pp. 1929–1934. IEEE, Piscataway (1989)

    Google Scholar 

  23. Bron, C.: Merge sort algorithm [m1]. Commun. ACM 15(5), 357–358 (1972)

    Article  Google Scholar 

  24. Burkard, R.E., Derigs, U.: Assignment and matching problems: solution methods with FORTRAN-programs. LN in Economics and Mathematical Systems, vol. 184. Springer, Berlin (1980)

    Google Scholar 

  25. Chanas, S., Kobylański, P.: A New Heuristic Algorithm Solving the Linear Ordering Problem. Comput. Optim. Appl. 6(2), 191–205 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Charon, I., Guénoche, A., Hudry, O., Woirgard, F.: New results on the computation of median orders. Discrete Math. 165–166, 139–153 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Charon, I., Hudry, O.: The noising method: a new method for combinatorial optimization. Oper. Res. Lett. 14(3), 133–137 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Charon, I., Hudry, O.: A branch-and-bound algorithm to solve the linear ordering problem for weighted tournaments. Discrete Appl. Math. 154(15), 2097–2116 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Cheung, L.K., Kuh, E.: The bordered triangular matrix and minimum essential sets of a digraph. IEEE Trans. Circuits Syst. 21(5), 633–639 (1974)

    Article  MathSciNet  Google Scholar 

  30. Chvatal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  31. Coleman, T., Wirth, A.: Ranking tournaments: Local search and a new algorithm. ACM J. Exp. Algorithm. 14(2.6), 1–22 (2009)

    Google Scholar 

  32. Cook, S.A.: A taxonomy of problems with fast parallel algorithms. Inform. Control 64(1–3), 2–22 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  33. Coppersmith, D., Fleischer, L.K., Rurda, A.: Ordering by weighted number of wins gives a good ranking for weighted tournaments. ACM Trans. Algorithms 6(3), 1–13 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Corbett, P.F.: Rotator graphs: an efficient topology for point-to-point multiprocessor networks. IEEE Trans. Parallel Distrib. Syst. 3(5), 622–626 (1992)

    Article  Google Scholar 

  35. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge, MA (2009)

    MATH  Google Scholar 

  36. Csardi, G., Nepusz, T.: The igraph software package for complex network research (2006)

    Google Scholar 

  37. de Souza, C.C., Keunings, R., Wolsey, L.A., Zone, O.: A new approach to minimising the frontwidth in finite element calculations. Comput. Methods Appl. Mech. Eng. 111(3–4), 323–334 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  38. Decani, J.S.: A branch and bound algorithm for maximum likelihood paired comparison ranking. Biometrika 59(1), 131–135 (1972)

    Article  MathSciNet  Google Scholar 

  39. Demetrescu, C., Finocchi, I.: Combinatorial algorithms for feedback problems in directed graphs. Inform. Proc. Lett. 86(3), 129–136 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Diaz, M., Richard, J., Courvoisier, M.: A note on minimal and quasi-minimal essential sets in complex directed graphs. IEEE Trans. Circuit Theory 19(5), 512–513 (1972)

    Article  MathSciNet  Google Scholar 

  41. Dom, M., Guo, J., Huffner, F., Niedermeier, R., Truss, A.: Fixed-parameter tractability results for feedback set problems in tournaments. In: CIAC: Italian Conference on Algorithms and Complexity, Lecture Notes in Computer Science, vol. 3998, pp. 320–331. Springer, Berlin-Heidelberg (2006)

    Google Scholar 

  42. Dom, M., Guo, J., Huffner, F., Niedermeier, R., Truss, A.: Fixed-parameter tractability results for feedback set problems in tournaments. J. Discrete Algorithms 8(1), 76–86 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Dorigo, M., Gambardella, L.M.: Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans. Evol. Comput. 1(1), 53–66 (1997)

    Article  Google Scholar 

  44. Du, D.Z., Hwang, F.K.: Generalized de Bruijn digraphs. Networks 18(1), 27–38 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  45. Dwork, C., Kumar, R., Naor, M., Sivakumar, D.: Rank aggregation methods for the web. In: WWW ’01: Proceedings of the 10th international conference on World Wide Web, pp. 613–622. Association for Computing Machinery (2001)

    Google Scholar 

  46. Eades, P., Lin, X.: A heuristic for the feedback arc set problem. Aust. J. Comb. 12, 15–25 (1995)

    MathSciNet  MATH  Google Scholar 

  47. Eades, P., Lin, X., Smyth, W.: A fast and effective heuristic for the feedback arc set problem. Inform. Proc. Lett. 47(6), 319–323 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  48. Edwards, C.S.: A branch and bound algorithm for the Koopmans–Beckmann quadratic assignment problem. In: Mathematical Programming Studies. Mathematical Programming Studies, vol. 13, pp. 35–52. Springer, Berlin-Heidelberg (1980)

    Google Scholar 

  49. Even, G., Naor, J.S., Rao, S., Schieber, B.: Divide-and-conquer approximation algorithms via spreading metrics. J. ACM 47(4), 585–616 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  50. Even, G., Naor, J.S., Schieber, B., Sudan, M.: Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20(2), 151–174 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  51. Feige, U.: Faster FAST (Feedback Arc Set in Tournaments) (2009)

    Google Scholar 

  52. Festa, P., Pardalos, P.M., Resende, M.G.C.: Feedback set problems. In: Handbook of Combinatorial Optimization, vol. A, pp. 209–258. Springer (1999)

    Google Scholar 

  53. Festa, P., Pardalos, P.M., Resende, M.G.C.: Algorithm 815: Fortran subroutines for computing approximate solutions of feedback set problems using grasp. ACM Trans. Math. Softw. 27(4), 456–464 (2001)

    Article  MATH  Google Scholar 

  54. Flood, M.M.: Exact and heuristic algorithms for the weighted feedback arc set problem: a special case of the skew-symmetric quadratic assignment problem. Networks 20(1), 1–23 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  55. Flueck, J.A., Korsh, J.F.: A branch search algorithm for maximum likelihood paired comparison ranking. Biometrika 61(3), 621–626 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  56. Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S.: Fast local search algorithm for weighted feedback arc set in tournaments. In: AAAI’10: Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence, vol. 24, pp. 65–70. AAAI Press (2010)

    Google Scholar 

  57. Frieze, A., Kannan, R.: Quick approximation to matrices and applications. Combinatorica 19(2), 175–220 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  58. Galinier, P., Lemamou, E., Bouzidi, M.W.: Applying local search to the feedback vertex set problem. J. Heuristics 19(5), 797–818 (2013)

    Article  Google Scholar 

  59. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco (1979)

    Google Scholar 

  60. Gillman, D.: A chernoff bound for random walks on expander graphs. SIAM J. Comput. 27(4), 1203–1220 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  61. Grotschel, M., Junger, M., Reinelt, G.: Acyclic subdigraphs and linear orderings: polytopes, facets, and a cutting plane algorithm. In: Graphs and Order. NATO ASI Series (Series C: Mathematical and Physical Sciences), vol. 147, pp. 217–264. Springer, Dordrecht (1985)

    Google Scholar 

  62. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing—STOC ’96, pp. 212–219. ACM Press (1996)

    Google Scholar 

  63. Guardabassi, G., Sangiovanni-Vincentelli, A.: A two levels algorithm for tearing. IEEE Trans. Circuits Syst. 23(12), 783–791 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  64. Gupta, S.: Feedback arc set problem in bipartite tournaments. Inform. Proc. Lett. 105(4), 150–154 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  65. Gupte, M., Shankar, P., Li, J., Muthukrishnan, S., Iftode, L.: Finding hierarchy in directed online social networks. In: Proceedings of the 20th International Conference on World Wide Web, pp. 557–566. ACM, New York (2011)

    Google Scholar 

  66. Gurevich, Y., Shelah, S.: Expected computation time for Hamiltonian path problem. SIAM J. Comput. 16(3), 486–502 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  67. Gurobi Optimization, LLC.: Gurobi Optimizer

    Google Scholar 

  68. Hagberg, A.A., Schult, D.A., Swart, P.J.: Exploring network structure, dynamics, and function using networkx. In: In Proceedings of the 7th Python in Science Conference (SciPy2008), pp. 11–15 (2008)

    Google Scholar 

  69. Hassin, R., Rubinstein, S.: Approximations for the maximum acyclic subgraph problem. Inform. Proc. Lett. 51(3), 133–140 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  70. Hecht, M., Gonciarz, K., Horvát, S.: Tight localizations of feedback sets. ACM J. Exp. Algorithmics 26, 1–19 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  71. Hecht, M.S., Ullman, J.D.: Flow graph reducibility. SIAM J. Comput. 1(2), 188–202 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  72. Hecht, M.S., Ullman, J.D.: Characterizations of reducible flow graphs. J. ACM 21(3), 367–375 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  73. Hoare, C.A.R.: Algorithm 64: quicksort. Commun. ACM 4(7), 321 (1961)

    Google Scholar 

  74. Horvat, S.: IGraph/M (2020).

    Google Scholar 

  75. Itoh, M: A design for directed graphs with minimum diameter. IEEE Trans. Comput. C-32(8), 782–784 (1983)

    Article  MATH  Google Scholar 

  76. Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  77. Johnson, D.B.: Finding all the elementary circuits of a directed graph. SIAM J. Comput. 4(1), 77–84 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  78. Kaas, R.: A branch and bound algorithm for the acyclic subgraph problem. Eur. J. Oper. Res. 8(4), 355–362 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  79. Kaku, B.K., Thompson, G.L., Baybars, I.: A heuristic method for the multi-story layout problem. Eur. J. Oper. Res. 37(3), 384–397 (1988)

    Article  MATH  Google Scholar 

  80. Karpinski, M., Schudy, W.: Faster algorithms for feedback arc set tournament, Kemeny rank aggregation and betweenness tournament. In: Algorithms and Computation. Lecture Notes in Computer Science, vol. 6506, pp. 3–14. Springer, Berlin (2010)

    Google Scholar 

  81. Kautz, W.H.: Unit-distance error-checking codes. IEEE Trans. Electron. Comput. EC-7(2), 179–180 (1958)

    Article  Google Scholar 

  82. Kendall, M.G.: Further contributions to the theory of paired comparisons. Biometrics 11(1), 43 (1955)

    Article  MathSciNet  Google Scholar 

  83. Kenyon-Mathieu, C., Schudy, W.: How to rank with few errors. In: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing—STOC ’07, pp. 95–103. ACM Press (2007)

    Google Scholar 

  84. Klein, P., Stein, C., Tardos, É.: Leighton-rao might be practical: faster approximation algorithms for concurrent flow with uniform capacities. In: Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing—STOC ’90, pp. 310–321. ACM Press (1990)

    Google Scholar 

  85. Knuth, D.E.: Sorting and Searching. The Art of Computer Programming, vol. 3, 2nd edn. Addison-Wesley Professional, Reading, MA (1998)

    Google Scholar 

  86. Koehler, H.: A contraction algorithm for finding minimal feedback sets. In: ACSC ’05: Proceedings of the Twenty-eighth Australasian Conference on Computer Science, vol. 38, pp. 165–173. Australian Computer Society (2005)

    Google Scholar 

  87. Kudelić, R.: Monte-carlo randomized algorithm for minimum feedback arc set. Appl. Soft Comput. 41, 235–246 (2016)

    Article  Google Scholar 

  88. Kudelić, R., Ivković, N.: Ant inspired monte carlo algorithm for minimum feedback arc set. Expert Syst. Appl. 122, 108–117 (2019)

    Article  Google Scholar 

  89. Kudelić, R., Rabuzin, K.: Dealing with intractability of information system subsystems development order via control flow graph reducibility. In: Proceedings of the 2020 3rd International Conference on Electronics and Electrical Engineering Technology, pp. 62–68. ACM, New York (2020)

    Google Scholar 

  90. Kuo, C.J., Hsu, C.C., Lin, H.R., Chen, D.R.: Minimum feedback arc sets in rotator graphs. In: The 26th Workshop on Combinatorial Mathematics and Computation Theory, pp. 95–101 (2009)

    Google Scholar 

  91. Lee, W., Rudd, D.F.: On the ordering of recycle calculations. AIChE J. 12(6), 1184–1190 (1966)

    Article  Google Scholar 

  92. Leighton, T., Rao, S.: An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In: [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science, pp. 422–431. IEEE, Piscataway (1988)

    Google Scholar 

  93. Lempel, A., Cederbaum, I.: Minimum feedback arc and vertex sets of a directed graph. IEEE Trans. Circuit Theory 13(4), 399–403 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  94. Levy, H., Low, D.W.: A contraction algorithm for finding small cycle cutsets. J. Algorithms 9(4), 470–493 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  95. Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  96. Luby, M., Nisan, N.: A parallel approximation algorithm for positive linear programming. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing—STOC ’93, pp. 448–457. ACM Press (1993)

    Google Scholar 

  97. Mathieu, C., Schudy, W.: How to Rank with Fewer Errors: A PTAS for Feedback Arc Set in Tournaments. Preliminary version in STOC 2007 (Kenyon-Mathieu and Schudy 2007)

    Google Scholar 

  98. Mathieu, C., Schudy, W.: Yet another algorithm for dense max cut: go greedy. In: SODA ’08: Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 176–182. Society for Industrial and Applied Mathematics (2008)

    Google Scholar 

  99. Matuszewski, C., Schonfeld, R., Molitor, P.: Using sifting for k-layer straightline crossing minimization. In: Graph Drawing. Lecture Notes in Computer Science, vol. 1731, pp. 217–224. Springer, Berlin (1999)

    Google Scholar 

  100. Mezard, M., Montanari, A.: Information, Physics, and Computation. Oxford University Press, Oxford (2009)

    Book  MATH  Google Scholar 

  101. Muñoz, X., Unger, W., Vrt’o, I.: One sided crossing minimization is NP-hard for sparse graphs. In: GD: International Symposium on Graph Drawing. Lecture Notes in Computer Science, vol. 2265, pp. 115–123. Springer, Berlin (2002)

    Google Scholar 

  102. Noughabi, H.A., Baghbani, F.G.: An efficient genetic algorithm for the feedback set problems. In: 2014 Iranian Conference on Intelligent Systems (ICIS). IEEE, Piscataway (2014)

    Google Scholar 

  103. Eades, P., Smyth, W.F., Lin, X.: Heuristics for the Feedback Arc Set Problem. Techreport 1, School of Computing Science, Curtin University of Technology, Perth, Western Australia (1989)

    Google Scholar 

  104. Pardalos, P.M., Qian, T., Resende, M.G.: A greedy randomized adaptive search procedure for the feedback vertex set problem. J. Comb. Optim. 2(4), 399–412 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  105. Park, S., Akers, S.: An efficient method for finding a minimal feedback arc set in directed graphs. In: [Proceedings] 1992 IEEE International Symposium on Circuits and Systems, pp. 1863–1866. IEEE, Piscataway (1992)

    Google Scholar 

  106. Phillips, J.P.N.: A Procedure for determining Slater’s i and all nearest adjoining orders. Br. J. Math. Stat. Psychol. 20(2), 217–225 (1967)

    Article  Google Scholar 

  107. Phillips, J.P.N.: A Further procedure for determining Slater’s i and all nearest adjoining orders. Br. J. Math. Stat. Psychol. 22(1), 97–101 (1969)

    Article  Google Scholar 

  108. Pho, T.K., Lapidus, L.: Topics in computer-aided design: part I. An optimum tearing algorithm for recycle systems. AIChE J. 19(6), 1170–1181 (1973)

    Google Scholar 

  109. Raghavan, P.: Probabilistic construction of deterministic algorithms: approximating packing integer programs. J. Comput. Syst. Sci. 37(2), 130–143 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  110. Ramachandran, V.: Finding a minimum feedback arc set in reducible flow graphs. J. Algorithms 9(3), 299–313 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  111. Ramachandran, V.: A minimax arc theorem for reducible flow graphs. SIAM J. Discrete Math. 3(4), 554–560 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  112. Raman, V., Saurabh, S.: Improved parameterized algorithms for feedback set problems in weighted tournaments. In: Parameterized and Exact Computation. Lecture Notes in Computer Science, vol. 3162, pp. 260–270. Springer, Berlin (2004)

    Google Scholar 

  113. Raman, V., Saurabh, S.: Parameterized algorithms for feedback set problems and their duals in tournaments. Theor. Comput. Sci. 351(3), 446–458 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  114. Raman, V., Saurabh, S., Sikdar, S.: Improved exact exponential algorithms for vertex bipartization and other problems. In: ICTCS: Italian Conference on Theoretical Computer Science. LNCS, vol. 3701, pp. 375–389. Springer, Berlin (2005)

    Google Scholar 

  115. Raman, V., Saurabh, S., Sikdar, S.: Efficient exact algorithms through enumerating maximal independent sets and other techniques. Theory Comput. Syst. 41(3), 563–587 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  116. Rosen, B.K.: Robust linear algorithms for cutsets. J. Algorithms 3(3), 205–217 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  117. Saab, Y.: A fast and effective algorithm for the feedback arc set problem. J. Heuristics 7(3), 235–250 (2001)

    Article  MATH  Google Scholar 

  118. Saab, Y.G.: A fast and robust network bisection algorithm. IEEE Trans. Comput. 44(7), 903–913 (1995)

    Article  MATH  Google Scholar 

  119. Schwikowski, B., Speckenmeyer, E.: On enumerating all minimal solutions of feedback problems. Discrete Appl. Math. 117(1-3), 253–265 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  120. Sedgewick, R., Wayne, K.: Algorithms, 4th edn. Addison-Wesley, Reading, MA (2011)

    Google Scholar 

  121. Seymour, P.D.: Packing directed circuits fractionally. Combinatorica 15(2), 281–288 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  122. Shamir, A.: A linear time algorithm for finding minimum cutsets in reducible graphs. SIAM J. Comput. 8(4), 645–655 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  123. Simpson, M., Srinivasan, V., Thomo, A.: Efficient computation of feedback arc set at web-scale. Proc. VLDB Endowment 10(3), 133–144 (2016)

    Article  Google Scholar 

  124. Smith, G., Walford, R.: The identification of a minimal feedback vertex set of a directed graph. IEEE Trans. Circuits Syst. 22(1), 9–15 (1975)

    Article  MathSciNet  Google Scholar 

  125. Spencer, J.: Ten Lectures on the Probabilistic Method, 2nd edn. Society for Industrial and Applied Mathematics (1994). (1987 edition not acquirable)

    Google Scholar 

  126. Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  127. Tarjan, R.E.: Testing flow graph reducibility. J. Comput. Syst. Sci. 9(3), 355–365 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  128. Tucker, A.W.: On directed graphs and integer programs. In: Symposium on Combinatorial Problems. Princeton University, Princeton (1960)

    Google Scholar 

  129. Waggener, W.M.: Pulse Code Modulation Techniques, 1st edn. Springer (1995)

    Google Scholar 

  130. Wang, C.C., Lloyd, E.L., Soffa, M.L.: Feedback vertex sets and cyclically reducible graphs. J. ACM 32(2), 296–313 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  131. Xiao, H.: Packing feedback arc sets in reducible flow graphs. J. Comb. Optim. 32(3), 951–959 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  132. Xu, Y.Z., Zhou, H.J.: Optimal segmentation of directed graph and the minimum number of feedback arcs. J. Stat. Phys. 169(1), 187–202 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  133. Younger, D.: Minimum feedback arc sets for a directed graph. IEEE Trans. Circuit Theory 10(2), 238–245 (1963)

    Article  Google Scholar 

  134. Zhao, J.H., Zhou, H.J.: Optimal Disruption of Complex Networks (2016)

    Google Scholar 

  135. van Zuylen, A.: Linear programming based approximation algorithms for feedback set problems in bipartite tournaments. Theor. Comput. Sci. 412(23), 2556–2561 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  136. van Zuylen, A., Hegde, R., Jain, K., Williamson, D.P.: Deterministic pivoting algorithms for constrained ranking and clustering problems. In: SODA ’07: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 405–414. Society for Industrial and Applied Mathematics (2007)

    Google Scholar 

  137. van Zuylen, A., Williamson, D.P.: Deterministic algorithms for rank aggregation and other ranking and clustering problems. In: Approximation and Online Algorithms. LNCS, vol. 4927, pp. 260–273. Springer, Berlin (2007)

    Google Scholar 

  138. van Zuylen, A., Williamson, D.P.: Deterministic pivoting algorithms for constrained ranking and clustering problems. Math. Oper. Res. 34(3), 594–620 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Faculty of Organization and Information Science, Theoretical and Applied Foundations of Information Science, University of Zagreb, Varaždin, Croatia

    Robert Kudelić

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Kudelić, R. (2022). Papers and Algorithms. In: Feedback Arc Set. SpringerBriefs in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-031-10515-9_3

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