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A unified algorithm for degree bounded survivable network design

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Abstract

We present an approximation algorithm for the minimum bounded degree Steiner network problem that returns a Steiner network of cost at most two times the optimal and the degree on each vertex \(v\) is at most \(\min \{b_v + 3r_{\mathrm{max}}, 2b_v+2\}\), where \(r_{\mathrm{max}}\) is the maximum connectivity requirement and \(b_v\) is the given degree bound on \(v\). This unifies, simplifies, and improves the previous results for this problem.

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Notes

  1. For instance, it was not known how to combine the results in [14, 15] to obtain a \((2, \min \{b_v+6r_{\mathrm{max}}+3, 2b_v+2\})\)-approximation algorithm for the problem.

  2. In particular, the analysis of the \((2,b_v+3)\) result is significantly simpler than that in [14].

References

  1. Bansal, N., Khandekar, R., Nagarajan, V.: Additive guarantees for degree bounded directed network design. SIAM J. Comput. 29, 1413–1431 (2009)

    MathSciNet  Google Scholar 

  2. Cheriyan, J., Vegh, L.: Approximating minimum-cost \(k\)-node connected subgraphs via independence-free graphs. In: Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2013)

  3. Cheriyan, J., Vempala, S., Vetta, A.: Network design via iterative rounding of setpair relaxations. Combinatorica 26(3), 255–275 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fleischer, L., Jain, K., Williamson, D.P.: Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems. J. Comput. Syst. Sci. 72(5), 838–867 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fukunaga, T., Ravi, R.: Iterative rounding approximation algorithms for degree-bounded node-connectivity network design. In: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 263–272 (2012)

  6. Fürer, M., Raghavachari, B.: Approximating the minimum-degree Steiner tree to within one of optimal. J. Algorithms 17(3), 409–423 (1994)

    Article  MathSciNet  Google Scholar 

  7. Gabow, H.: On the \(L_\infty \)-norm of extreme points for crossing supermodular directed network LPs. In: Proceedings of the 11th International Conference on Integer Programming and Combinatorial Optimization (IPCO), pp. 392–406 (2005)

  8. Goemans, M.X.: Minimum bounded-degree spanning trees. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 273–282 (2006)

  9. Jain, K.: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21(1), 39–60 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Khandekar, R., Kortsarz, G., Nutov, Z.: On some network design problems with degree constraints. J. Comput. Syst. Sci. 79(5), 725–736 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Király, T., Lau, L.C., Singh, M.: Degree bounded matroids and submodular flows. In: Proceedings of the 13th International Conference on Integer Programming and Combinatorial Optimization (IPCO), pp. 259–272 (2008)

  12. Lau, L.C., Naor, S., Salavatipour, M., Singh, M.: Survivable network design with degree or order constraints. SIAM J. Comput. 39(3), 1062–1087 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lau, L.C., Ravi, R., Singh, M.: Iterative Methods in Combinatorial Optimization. Cambridge University Press, New York (2011)

    Book  MATH  Google Scholar 

  14. Lau, L.C., Singh, M.: Additive approximation for bounded degree survivable network design. SIAM J. Comput. 42(6), 2217–2242 (2013)

  15. Louis, A., Vishnoi, N.K.: Improved algorithm for degree bounded survivable network design problem. In: Proceedings of the 12th Scandinavian Symposium and Workshops on Algorithm Theory, pp. 408–419 (2010)

  16. Nutov, Z.: Degree-constrained node-connectivity. In: Proceedings of the 10th Latin American Symposium on Theoretical Informatics (LATIN), pp. 582–593 (2012)

  17. Singh, M., Lau, L.C.: Approximating minimum bounded degree spanning trees to within one of optimal. In: Proceedings of the 39th ACM Symposium on Theory of Computing (STOC), pp. 661–670 (2007)

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Acknowledgments

We thank the anonymous reviewers for comments that improve the presentation of the paper. This research is partially supported by HK RGC Grant 2150701.

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  1. Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong

    Lap Chi Lau & Hong Zhou

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  1. Lap Chi Lau
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  2. Hong Zhou
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Correspondence to Hong Zhou.

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Lau, L.C., Zhou, H. A unified algorithm for degree bounded survivable network design. Math. Program. 154, 515–532 (2015). https://doi.org/10.1007/s10107-015-0858-5

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  • DOI: https://doi.org/10.1007/s10107-015-0858-5

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