COCOON 2012: Computing and Combinatorics pp 169-180

# 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
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

## Abstract

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.

## Keywords

set cover minimum degree hypergraphs 2-satisfiability algorithms

## References

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)
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)
3. 3.
Caprara, A., Toth, P., Fischetti, M.: Algorithms for the set covering problem. Annals of Operations Research 98, 353–371 (2000)
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)
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)
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)
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)
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)
15. 15.
Tarjan, R.: Depth-first search and linear graph algorithms. SIAM Journal on Computing 1(2), 146–160 (1972)
16. 16.
Veinott, A.F., Wagner, H.M.: Optimal capacity scheduling. Operations Research 10(4), 518–532 (1962)