, Volume 76, Issue 2, pp 320–343 | Cite as

Algorithms and Kernels for Feedback Set Problems in Generalizations of Tournaments

  • Jørgen Bang-Jensen
  • Alessandro Maddaloni
  • Saket Saurabh


In the Directed Feedback Arc (Vertex) Set problem, we are given a digraph D with vertex set V(D) and arcs set A(D) and a positive integer k, and the question is whether there is a subset X of arcs (vertices) of size at most k such that the digraph obtained after deleting X from D is an acyclic digraph. In this paper we study these two problems in the realm of parametrized and kernelization complexity. More precisely, for these problems we give polynomial time algorithms, known as kernelization algorithms, on several digraph classes that given an instance (Dk) of the problem returns an equivalent instance \((D',k')\) such that the size of \(D'\) and \(k'\) is at most \(k^{O(1)}\). We extend previous results for Directed Feedback Arc (Vertex) Set on tournaments to much larger and well studied classes of digraphs. Specifically we obtain polynomial kernels for k-FVS on digraphs with bounded independence number, locally semicomplete digraphs and some totally \(\Phi \)-decomposable digraphs, including quasi-transitive digraphs. We also obtain polynomial kernels for k-FAS on some totally \(\Phi \)-decomposable digraphs, including quasi-transitive digraphs. Finally, we design a subexponential algorithm for k-FAS running in time \(2^{O(\sqrt{k} (\log k)^c)}n^d\) for constants cd. on locally semicomplete digraphs.


Parameterized complexity Kernels Feedback vertex set Feedback arc set Decomposable digraph Bounded independence number Locally semicomplete digraph Quasi-transitive digraph 


  1. 1.
    Abu-Khzam, F.N.: A kernelization algorithm for d-hitting set. J. Comput. Syst. Sci. 76(7), 524–531 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: ranking and clustering. In: Proc. STOC’05: the 37th Annual ACM Symp. on Theory of Computing, pp. 684–693. ACM Press, New York (2005)Google Scholar
  3. 3.
    Alon, N.: Ranking tournaments. SIAM J. Discrete Math. 20, 137–142 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alon, N., Lokshtanov, D., Saurabh, S.: Fast FAST. In: Proc. 36. ICALP, Lecture Notes in Computer Science 5555, pp. 49–58. Springer, Berlin (2009)Google Scholar
  5. 5.
    Bang-Jensen, J.: Locally semicomplete digraphs: a generalization of tournaments. J. Graph Theory 14(3), 371–390 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications, 2nd edn. Springer, London (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bang-Jensen, J., Huang, J.: Quasi-transitive digraphs. J. Graph Theory 20(2), 141–161 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bang-Jensen, J., Maddaloni, A.: Arc-disjoint paths in decomposable digraphs. J. Graph Theory 77, 89–110 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bang-Jensen, J., Thomassen, C.: A polynomial algorithm for the 2-path problem for semicomplete digraphs. SIAM J. Discrete Math. 5, 366–376 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Charbit, P., Thomassé, S., Yeo, A.: The minimum feedback arc set problem is NP-hard for tournaments. Comb. Probab. Comput. 16, 1–4 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55(5) (2008)Google Scholar
  14. 14.
    Chudnovsky, M., Fradkin, A., Seymour, P.: Tournament immersion and cutwidth. J. Comb. Theory Ser. B 102, 93–101 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Conitzer, V.: Computing Slater rankings using similarities among candidates. In: 21st National Conference on Artificial Intelligence (AAAI-06), pp. 613–619 (2006)Google Scholar
  16. 16.
    Crespelle, C., Paul, C.: Fully-dynamic recognition algorithm and certificate for directed cographs. Discrete Appl. Math. 154(12), 1722–1741 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: FOCS, pp. 150–159 (2011)Google Scholar
  18. 18.
    Dom, M., Guo, J., Hüffner, F., Niedermeier, R., Truß, A.: Fixed-parameter tractability results for feedback set problems in tournaments. J. Discrete Algorithms 8(1), 76–86 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Erdős, P., Rado, R.: Intersection theorems for systems of sets. J. Lond. Math. Soc. 35, 85–90 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Even, G., Naor, J., Schieber, B., Sudan, M.: Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20(2), 151–174 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Feige, U.: Faster FAST(Feedback Arc Set in Tournaments). CoRR, abs/0911.5094 (2009)Google Scholar
  22. 22.
    Flum, J., Grohe, M.: Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series). Springer, New York (2006)Google Scholar
  23. 23.
    Fomin, F.V., Pilipczuk, M.: Jungles, bundles and fixed-parameter tractability. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, pp. 396–413 (2013)Google Scholar
  24. 24.
    Fomin, F.V., Pilipczuk, M.: Subexponential parameterized algorithm for computing the cutwidth of a semi-complete digraph. In: Proceedings of ESA 2013, Lecture Notes in Computer Science, vol. 8125, pp. 505–516 (2013)Google Scholar
  25. 25.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Garey, M.R., Johnson, D.S.: Computers and intractability. W. H. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  27. 27.
    Ghouila-Houri, A.: Caractérisation des graphes non orientés dont on peut orienter les arětes de manière à obtenir le graphe d’une relation d’ordre. C. R. Acad. Sci. Paris 254, 1370–1371 (1962)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Guo, Y.: Locally semicomplete digraphs. PhD thesis, RWTH Aachen, Germany (1995)Google Scholar
  29. 29.
    Gutin, G.: Polynomial algorithms for finding Hamiltonian paths and cycles in quasi-transitive digraphs. Australas. J. Comb. 10, 231–236 (1994)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Gutin, G., Yeo, A.: Some parameterized problems on digraphs. Comput. J. 51, 363–371 (2008)CrossRefGoogle Scholar
  31. 31.
    Hsiao, S.-Y.: Fixed-parameter complexity of feedback vertex set in bipartite tournaments. ISAAC 7074, 344–353 (2011)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Huang, J.: Tournament-like oriented graphs. Ph.D. thesis, School of Computing Science, Simon Fraser University, Canada (1992)Google Scholar
  33. 33.
    Huang, J.: On the structure of local tournaments. J. Comb. Theory Ser. B 63(2), 200–221 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Huang, J.: Which digraphs are round? Australas. J. Comb. 19, 203–208 (1999)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of computer computations (Proc. Symp., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), pp. 85–103. Plenum (1972)Google Scholar
  36. 36.
    Karpinski, M., Schudy, W.: Faster algorithms for feedback arc set tournament, Kemeny rank aggregation and betweenness tournament. ISAAC 1(6506), 3–14 (2010)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Kociumaka, T., Pilipczuk, M.: Faster deterministic feedback vertex set. Inf. Process. Lett. 114, 556–560 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Lokshtanov, D., Misra, N., Saurabh, S.: The multivariate algorithmic revolution and beyond. Chapter Kernelization—preprocessing with a guarantee, volume 7370 of Lecture Notes in Computer Science, pp. 129–161 (2012)Google Scholar
  39. 39.
    Misra, P., Raman, V., Ramanujan, M.S., Saurabh, S.: A polynomial kernel for feedback arc set on bipartite tournaments. Theory Comput. Syst. 53, 609–620 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Raman, V., Saurabh, S.: Parameterized algorithms for feedback set problems and their duals in tournaments. Theor. Comput. Sci 351(3), 446–458 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Skrien, D.J.: A relationship between triangulated graphs, comparability graphs, proper interval graphs, proper circular-arc graphs, and nested interval graphs. J. Graph Theory 6(3), 309–316 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Thomassé, S.: A \(4k^2\) kernel for feedback vertex set. ACM Trans. Algorithms 6, 8 (2010). Article 32Google Scholar
  43. 43.
    Xiao, M., Guo, J.: A quadratic vertex kernel for feedback arc set in bipartite tournaments. MFCS 7464, 825–835 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jørgen Bang-Jensen
    • 1
  • Alessandro Maddaloni
    • 2
  • Saket Saurabh
    • 3
  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark
  2. 2.TeCIP Institute, Scuola Superiore Sant’AnnaPisaItaly
  3. 3.The Institute of Mathematical SciencesChennaiIndia

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