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Smallest Bipartite Bridge-Connectivity Augmentation

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Abstract

This paper addresses two augmentation problems related to bipartite graphs. The first, a fundamental graph-theoretical problem, is how to add a set of edges with the smallest possible cardinality so that the resulting graph is 2-edge-connected, i.e., bridge-connected, and still bipartite. The second problem, which arises naturally from research on the security of statistical data, is how to add edges so that the resulting graph is simple and does not contain any bridges. In both cases, after adding edges, the graph can be either a simple graph or, if necessary, a multi-graph. Our approach then determines whether or not such an augmentation is possible.

We propose a number of simple linear-time algorithms to solve both problems. Given the well-known bridge-block data structure for an input graph, the algorithms run in O(log n) parallel time on an EREW PRAM using a linear number of processors, where n is the number of vertices in the input graph. We note that there is already a polynomial time algorithm that solves the first augmentation problem related to graphs with a given general partition constraint in O(n(m+nlog n)log n) time, where m is the number of distinct edges in the input graph. We are unaware of any results for the second problem.

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References

  1. Adam, N.R., Wortmann, J.C.: Security-control methods for statistical database: a comparative study. ACM Comput. Surv. 21, 515–556 (1989)

    Article  Google Scholar 

  2. Chin, F.Y., Özsoyoǧlu, G.: Auditing and inference control in statistical databases. IEEE Trans. Softw. Eng. 8, 574–582 (1982)

    Article  Google Scholar 

  3. Chong, K.W., Han, Y., Lam, T.W.: Concurrent threads and optimal parallel minimum spanning trees algorithm. J. ACM 48(2), 297–323 (2001)

    MATH  MathSciNet  Google Scholar 

  4. Cole, R., Vishkin, U.: Approximate parallel scheduling. Part II: Applications to logarithmic-time optimal graph algorithms. Inf. Comput. 92, 1–47 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cox, L.H.: Suppression methodology and statistical disclosure control. J. Am. Stat. Assoc. 75, 377–385 (1980)

    Article  MATH  Google Scholar 

  6. Denning, D.E., Schlörer, J.: Inference controls for statistical databases. IEEE Comput. 16, 69–82 (1983)

    Google Scholar 

  7. Eswaran, K.P., Tarjan, R.E.: Augmentation problems. SIAM J. Comput. 5, 653–665 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  8. Frank, A.: Connectivity augmentation problems in network design. In: Birge, J.R., Murty, K.G. (eds.) Mathematical Programming: State of the Art 1994, pp. 34–63. University of Michigan Press, Ann Arbor (1994)

    Google Scholar 

  9. Gusfield, D.: A graph theoretic approach to statistical data security. SIAM J. Comput. 17, 552–571 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  10. Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)

    Google Scholar 

  11. Hsu, T.-s.: Graph Augmentation and Related Problems: Theory and Practice. Ph.D. Thesis, University of Texas at Austin (1993)

  12. Hsu, T.-s.: Undirected vertex-connectivity structure and smallest four-vertex-connectivity augmentation (extended abstract). In: Staples, J. (ed.) Proceedings of the 6th International Symposium on Algorithms and Computation. Lecture Notes in Computer Science, vol. 1004, pp. 274–283. Springer, New York (1995)

    Google Scholar 

  13. Hsu, T.-s.: On four-connecting a triconnected graph. J. Algorithms 35, 202–234 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hsu, T.-s.: Simpler and faster vertex-connectivity augmentation algorithms (extended abstract). In: Paterson, M. (ed.) Proceedings of the 8th European Symposium on Algorithms. Lecture Notes in Computer Science, vol. 1879, pp. 278–289. Springer, New York (2000)

    Google Scholar 

  15. Hsu, T.-s.: Simpler and faster biconnectivity augmentation. J. Algorithms 45(1), 55–71 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hsu, T.-s., Kao, M.Y.: Security problems for statistical databases with general cell suppressions. In: Proceedings of the 9th International Conference on Scientific and Statistical Database Management, pp. 155–164 (1997)

  17. Hsu, T.-s., Kao, M.Y.: Optimal augmentation for bipartite componentwise biconnectivity in linear time. SIAM J. Discrete Math. 19(2), 345–362 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hsu, T.-s., Ramachandran, V.: On finding a smallest augmentation to biconnect a graph. SIAM J. Comput. 22, 889–912 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jensen, J.B., Gabow, H.N., Jordan, T., Szigeti, Z.: Edge-connectivity augmentation with partition constraints. SIAM J. Discrete Math. 12, 160–207 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kant, G.: Algorithms for Drawing Planar Graphs. Ph.D. Thesis, Utrecht University, the Netherlands (1993)

  21. Kao, M.Y.: Linear-time optimal augmentation for componentwise bipartite-completeness of graphs. Inf. Process. Lett. 59–63 (1995)

  22. Kao, M.Y.: Data security equals graph connectivity. SIAM J. Discrete Math. 9, 87–100 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kao, M.Y.: Total protection of analytic-invariant information in cross-tabulated tables. SIAM J. Comput. 26, 231–242 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kelly, J.P., Golden, B.L., Assad, A.A.: Cell suppression: disclosure protection for sensitive tabular data. Networks 22, 397–417 (1992)

    Article  MATH  Google Scholar 

  25. Malvestuto, F.M., Moscarini, M.: Censoring statistical tables to protect sensitive information: easy and hard problems. In: Proceedings of the 8th International Conference on Scientific and Statistical Database Management, pp. 12–21 (1996)

  26. Malvestuto, F.M., Moscarini, M.: Suppressing marginal totals from a two-dimensional table to protect sensitive information. Stat. Comput. 7, 101–114 (1997)

    Article  Google Scholar 

  27. Malvestuto, F.M., Moscarini, M., Rafanelli, M.: Suppressing marginal cells to protect sensitive information in a two-dimensional statistical table. In: Proceedings of the 10th ACM SIGACT-SIGMOD-SIGACT Symposium on Principles of Database Systems, pp. 252–258 (1991)

  28. Nagamochi, H.: Recent development of graph connectivity augmentation algorithms. IEICE Trans. Inf. Syst. E83-D, 372–383 (2000)

    Google Scholar 

  29. Ramachandran, V.: Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. In: Reif, J.H. (ed.) Synthesis of Parallel Algorithms, pp. 275–340. Morgan Kaufmann, San Mateo (1993)

    Google Scholar 

  30. Rosenthal, A., Goldner, A.: Smallest augmentations to biconnect a graph. SIAM J. Comput. 6, 55–66 (1977)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Tarjan, R.E., Vishkin, U.: An efficient parallel biconnectivity algorithm. SIAM J. Comput. 14, 862–874 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  33. Watanabe, T., Nakamura, A.: A minimum 3-connectivity augmentation of a graph. J. Comput. Syst. Sci. 46, 91–128 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. EECS Rm 730, Department of Computer Science, National Tsing-Hua University, 101, Section 2, Kuang-Fu Road, Hsinchu, 30013, Taiwan

    Pei-Chi Huang, Hsin-Wen Wei, Wan-Chen Lu & Wei-Kuan Shih

  2. Institute of Information Science, Academia Sinica, Nankang, Taipei, Taiwan

    Tsan-sheng Hsu

Authors
  1. Pei-Chi Huang
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  2. Hsin-Wen Wei
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  3. Wan-Chen Lu
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  4. Wei-Kuan Shih
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  5. Tsan-sheng Hsu
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Correspondence to Hsin-Wen Wei.

Additional information

H.-W. Wei, W.-C. Lu and T.-s. Hsu research supported in part by NSC of Taiwan Grants 94-2213-E-001-014, 95-2221-E-001-004 and 96-2221-E-001-004.

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Huang, PC., Wei, HW., Lu, WC. et al. Smallest Bipartite Bridge-Connectivity Augmentation. Algorithmica 54, 353–378 (2009). https://doi.org/10.1007/s00453-007-9127-1

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  • DOI: https://doi.org/10.1007/s00453-007-9127-1

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