A Quadratic Vertex Kernel for Feedback Arc Set in Bipartite Tournaments

  • Mingyu Xiao
  • Jiong Guo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)

Abstract

The k-feedback arc set problem is to determine whether there is a set F of at most k arcs in a directed graph G such that the removal of F makes G acyclic. The k-feedback arc set problems in tournaments and bipartite tournaments (k-FAST and k-FASBT) have applications in ranking aggregation and have been extensively studied from the viewpoint of parameterized complexity. Recently, Misra et al. provide a problem kernel with O(k3) vertices for k-FASBT. Answering an open question by Misra et al., we improve the kernel bound to O(k2) vertices by introducing a new concept called “bimodule.”

Keywords

Kernelization Feedback arc set Bipartite tournament Graph algorithms Parameterized algorithms 

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References

  1. 1.
    Abu-Khzam, F.N.: A kernelization algorithm for d-hitting set. Journal of Computer and System Sciences 76(7), 524–531 (2010)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Alon, N.: Ranking tournaments. SIAM J. Discrete Math. 20(1), 137–142 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Alon, N., Lokshtanov, D., Saurabh, S.: Fast FAST. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 49–58. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Cai, M., Deng, M., Zang, W.: A min-max theorem on feedback vertex sets. Math. Oper. Res. 27(2), 361–371 (2002)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    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 (2009)CrossRefGoogle Scholar
  6. 6.
    Charbit, P., Thomassé, S., Yeo, A.: The minimum feedback arc set problem is NP-hard for tournaments. Combin. Probab. Comput. 16(1), 1–4 (2007)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    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)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Guo, J., Hüffner, F., Moser, H.: Feedback arc set in bipartite tournaments is NP-complete. Inf. Process. Lett. 102(2-3), 62–65 (2007)MATHCrossRefGoogle Scholar
  9. 9.
    Hsiao, S.-Y.: Fixed-Parameter Complexity of Feedback Vertex Set in Bipartite Tournaments. In: Asano, T., Nakano, S.-i., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 344–353. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Karpinski, M., Schudy, W.: Faster Algorithms for Feedback Arc Set Tournament, Kemeny Rank Aggregation and Betweenness Tournament. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part I. LNCS, vol. 6506, pp. 3–14. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Kemeny, J.: Mathematics without numbers. Daedalus 88, 571–591 (1959)Google Scholar
  12. 12.
    Kemeny, J., Snell, J.: Mathematical models in the social sciences. Blaisdell (1962)Google Scholar
  13. 13.
    Hsiao, S.-Y.: Fixed-Parameter Complexity of Feedback Vertex Set in Bipartite Tournaments. In: Asano, T., Nakano, S.-i., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 344–353. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Paul, C., Perez, A., Thomassé, S.: Conflict Packing Yields Linear Vertex-Kernels for k -FAST, k -dense RTI and a Related Problem. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 497–507. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Sanghvi, B., Koul, N., Honavar, V.: Identifying and Eliminating Inconsistencies in Mappings across Hierarchical Ontologies. In: Meersman, R., Dillon, T., Herrero, P. (eds.) OTM 2010. LNCS, vol. 6427, pp. 999–1008. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Speckenmeyer, E.: On Feedback Problems in Digraphs. In: Nagl, M. (ed.) WG 1989. LNCS, vol. 411, pp. 218–231. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  17. 17.
    Sasatte, P.: Improved FPT algorithm for feedback vertex set problem in bipartite tournament. Information Processing Letters 105(3), 79–82 (2008)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Wahlström, M.: Algorithms, Measures and Upper Bounds for Satisfiability and Related Problems. PhD thesis, Linköpings universitet (2007)Google Scholar
  19. 19.
    Raman, V., Saurabh, S.: Parameterized algorithms for feedback set problems and their duals in tournaments. Theor. Comput. Sci. 351(3), 446–458 (2006)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Mingyu Xiao
    • 1
  • Jiong Guo
    • 2
  1. 1.School of Computer Science and EngineeringUniversity of Electronic Science and Technology of ChinaChengduChina
  2. 2.Universität des SaarlandesSaarbrückenGermany

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