On the Minimum Degree Hypergraph Problem with Subset Size Two and the Red-Blue Set Cover Problem with the Consecutive Ones Property

  • Biing-Feng Wang
  • Chih-Hsuan Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)


Let S be a set and let C b (blue collection) and C r (red collection) be two collections of subsets of S. The MDH problem is to find a subset S′ ⊆ S such that S′ ∩ B ≠ ∅ for all B ∈ C b and |S′ ∩ R| ≤ k for all R ∈ C r , where k is a given non-negative integer. The RBSC problem is to find a subset S′ ⊆ S with S′ ∩ B ≠ ∅ for all B ∈ C b which minimizes |{R | R ∈ C r , S′ ∪ R ≠ ∅ }|. In this paper, improved algorithms are proposed for the MDH problem with k = 1 and all sets in C b having size two and the RBSC problem with C b  ∪ C r having the consecutive ones property. For the first problem, we give an optimal \(O(|S| + |C_{b}| + \sum_{R \in C_{r}} |R|)\)-time algorithm, improving the previous \(O(|S| + |C_{b}| + \sum_{R \in C_{r}} |R|^{2})\) bound by Dom et al. Our improvement is obtained by presenting a new representation of a dense directed graph, which may be of independent interest. For the second problem, we give an \(O(|C_{b}| + |C_{r}| \lg |S| + |S| \lg |S|)\)-time algorithm, improving the previous O(|C b ||S| + |C r ||S| + |S|2) bound by Chang et al.


set cover minimum degree hypergraphs 2-satisfiability algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aspvall, B., Plass, M.F., Tarjan, R.E.: A linear-time algorithm for testing the truth of certain quantified boolean formulas. Information Processing Letters 8(3), 121–123 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. Journal of Computer and System Sciences 13(3), 335–379 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Caprara, A., Toth, P., Fischetti, M.: Algorithms for the set covering problem. Annals of Operations Research 98, 353–371 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Carr, R.D., Doddi, S., Konjevod, G., Marathe, M.: On the red-blue set cover problem. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 345–353 (2000)Google Scholar
  5. 5.
    Chang, M.S., Chung, H.H., Lin, C.C.: An improved algorithm for the redvblue hitting set problem with the consecutive ones property. Information Processing Letters 110(20), 845–848 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw-Will (2001)Google Scholar
  7. 7.
    Dom, M., Guo, J., Niedermeier, R., Wernicke, S.: Red-blue covering problems and the consecutive ones property. Journal of Discrete Algorithms 6(3), 393–407 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Feder, T., Motwani, R., Zhu, A.: k-connected spanning subgraphs of low degree. Tech. Rep. TR06-041. Electronic Colloquium on Computational Complexity (2006)Google Scholar
  9. 9.
    Kuhn, F., von Rickenbach, P., Wattenhofer, R., Welzl, E., Zollinger, A.: Interference in Cellular Networks: The Minimum Membership Set Cover Problem. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 188–198. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Li, C.H., Ye, J.H., Wang, B.F.: A linear-time algorithm for the minimum degree hypergraph problem with the consecutive ones property (2012) (unpublished manuscript)Google Scholar
  11. 11.
    Mecke, S., Schöbel, A., Wagner, D.: Station location - complexity and approximation. In: 5th Workshop on Algorithmic Methods and Models for Optimization of Railways (2006)Google Scholar
  12. 12.
    Mecke, S., Wagner, D.: Solving Geometric Covering Problems by Data Reduction. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 760–771. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer-Verlag New York, Inc. (1985)Google Scholar
  14. 14.
    Ruf, N., Schobel, A.: Set covering with almost consecutive ones property. Discrete Optimization 1(2), 215–228 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Tarjan, R.: Depth-first search and linear graph algorithms. SIAM Journal on Computing 1(2), 146–160 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Veinott, A.F., Wagner, H.M.: Optimal capacity scheduling. Operations Research 10(4), 518–532 (1962)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Biing-Feng Wang
    • 1
  • Chih-Hsuan Li
    • 1
  1. 1.Department of Computer ScienceNational Tsing Hua UniversityHsinchuTaiwan, Republic of China

Personalised recommendations