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Approximation Algorithms for k-Connected Graph Factors

  • Bodo Manthey
  • Marten Waanders
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9499)

Abstract

Finding low-cost spanning subgraphs with given degree and connectivity requirements is a fundamental problem in the area of network design. We consider the problem of finding d-regular spanning subgraphs (or d-factors) of minimum weight with connectivity requirements. For the case of k-edge-connectedness, we present approximation algorithms that achieve constant approximation ratios for all \(d \ge 2 \cdot \lceil k/2 \rceil \). For the case of k-vertex-connectedness, we achieve constant approximation ratios for \(d \ge 2k-1\). Our algorithms also work for arbitrary degree sequences if the minimum degree is at least \(2 \cdot \lceil k/2 \rceil \) (for k-edge-connectivity) or \(2k-1\) (for k-vertex-connectivity).

Keywords

Approximation Algorithm Approximation Ratio Travel Salesman Problem Travel Salesman Problem Minimum Weight 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Chan, Y.H., Fung, W.S., Lau, L.C., Yung, C.K.: Degree bounded network design with metric costs. SIAM J. Comput. 40(4), 953–980 (2011)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Cheah, F., Corneil, D.G.: The complexity of regular subgraph recognition. Discrete Appl. Math. 27(1–2), 59–68 (1990)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cheriyan, J., Vempala, S., Vetta, A.: An approximation algorithm for the minimum-cost \(k\)-vertex connected subgraph. SIAM J. Comput. 32(4), 1050–1055 (2003)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cheriyan, J., Vetta, A.: Approximation algorithms for network design with metric costs. SIAM J. Discrete Math. 21(3), 612–636 (2007)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Cornelissen, K., Hoeksma, R., Manthey, B., Narayanaswamy, N.S., Rahul, C.S.: Approximability of connected factors. In: Kaklamanis, C., Pruhs, K. (eds.) WAOA 2013. LNCS, vol. 8447, pp. 120–131. Springer, Heidelberg (2014) Google Scholar
  6. 6.
    Czumaj, A., Lingas, A.: Minimum \(k\)-connected geometric networks. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms, pp. 536–539. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Fekete, S.P., Khuller, S., Klemmstein, M., Raghavachari, B., Young, N.E.: A network-flow technique for finding low-weight bounded-degree spanning trees. J. Algorithms 24(2), 310–324 (1997)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Fukunaga, T., Nagamochi, H.: Network design with edge-connectivity and degree constraints. Theor. Comput. Syst. 45(3), 512–532 (2009)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Fukunaga, T., Nagamochi, H.: Network design with weighted degree constraints. Discrete Optim. 7(4), 246–255 (2010)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fukunaga, T., Ravi, R.: Iterative rounding approximation algorithms for degree-bounded node-connectivity network design. In: Proceedings of the 53rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 263–272. IEEE Computer Society (2012)Google Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)MATHGoogle Scholar
  12. 12.
    Goemans, M.X., Bertsimas, D.: Survivable networks, linear programming relaxations and the parsimonious property. Math. Program. 60, 145–166 (1993)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Kammer, F., Täubig, H.: Connectivity. In: Brandes, U., Erlebach, T. (eds.) Network Analysis. LNCS, vol. 3418, pp. 143–177. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  14. 14.
    Khandekar, R., Kortsarz, G., Nutov, Z.: On some network design problems with degree constraints. J. Comput. Syst. Sci. 79(5), 725–736 (2013)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Khuller, S., Raghavachari, B.: Graph connectivity. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms, pp. 371–373. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Khuller, S., Vishkin, U.: Biconnectivity approximations and graph carvings. J. ACM 41(2), 214–235 (1994)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kortsarz, G., Nutov, Z.: Approximating node connectivity problems via set covers. Algorithmica 37(2), 75–92 (2003)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Lau, L.C., Naor, J., Salavatipour, M.R., Singh, M.: Survivable network design with degree or order constraints. SIAM J. Comput. 39(3), 1062–1087 (2009)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Lau, L.C., Singh, M.: Additive approximation for bounded degree survivable network design. SIAM J. Comput. 42(6), 2217–2242 (2013)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Lau, L.C., Zhou, H.: A unified algorithm for degree bounded survivable network design. In: Lee, J., Vygen, J. (eds.) IPCO 2014. LNCS, vol. 8494, pp. 369–380. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  21. 21.
    Lovász, L., Plummer, M.D.: Matching Theory, North-Holland Mathematics Studies, vol. 121. Elsevier (1986)Google Scholar
  22. 22.
    Singh, M., Lau, L.C.: Approximating minimum bounded degree spanning trees to within one of optimal. J. ACM 62(1), 1:1–1:19 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tutte, W.T.: A short proof of the factor theorem for finite graphs. Can. J. Math. 6, 347–352 (1954)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice-Hall, Upper Saddle River (2001)Google Scholar
  25. 25.
    Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, New York (2011)MATHCrossRefGoogle Scholar
  26. 26.
    Wolsey, L.A.: Heuristic analysis, linear programming and branch and bound. In: Rayward-Smith, V.J. (ed.) Combinatorial Optimization II, Mathematical Programming Studies, vol. 13, pp. 121–134. Springer, Heidelberg (1980)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University of TwenteEnschedeThe Netherlands

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