Approximability of Connected Factors

  • Kamiel Cornelissen
  • Ruben Hoeksma
  • Bodo Manthey
  • N. S. Narayanaswamy
  • C. S. Rahul
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8447)

Abstract

Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by Tutte’s reduction to the matching problem. By the same reduction, it is easy to find a minimal or maximal d-factor of a graph. However, if we require that the d-factor is connected, these problems become NP-hard – finding a minimal connected 2-factor is just the traveling salesman problem (TSP).

Given a complete graph with edge weights that satisfy the triangle inequality, we consider the problem of finding a minimal connected d-factor. We give a 3-approximation for all d and improve this to an (r + 1)-approximation for even d, where r is the approximation ratio of the TSP. This yields a 2.5-approximation for even d. The same algorithm yields an (r + 1)-approximation for the directed version of the problem, where r is the approximation ratio of the asymmetric TSP. We also show that none of these minimization problems can be approximated better than the corresponding TSP.

Finally, for the decision problem of deciding whether a given graph contains a connected d-factor, we extend known hardness results.

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References

  1. 1.
    Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An O(logn/loglogn)-approximation algorithm for the asymmetric traveling salesman problem. In: Proc. of the 21st Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 379–389. SIAM (2010)Google Scholar
  2. 2.
    Baburin, A.E., Gimadi, E.K.: Approximation algorithms for finding a maximum-weight spanning connected subgraph with given vertex degrees. In: Operations Research Proceedings 2004, pp. 343–351 (2005)Google Scholar
  3. 3.
    Baburin, A.E., Gimadi, E.K.: Polynomial algorithms for some hard problems of finding connected spanning subgraphs of extreme total edge weight. In: Operations Research Proceedings 2006, pp. 215–220 (2007)Google Scholar
  4. 4.
    Baburin, A.E., Gimadi, E.K.: An approximation algorithm for finding a d-regular spanning connected subgraph of maximum weight in a complete graph with random weights of edges. Journal of Applied and Industrial Mathematics 2(2), 155–166 (2008)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Chan, Y.H., Fung, W.S., Lau, L.C., Yung, C.K.: Degree bounded network design with metric costs. SIAM Journal on Computing 40(4), 953–980 (2011)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Cheah, F., Corneil, D.G.: The complexity of regular subgraph recognition. Discrete Applied Mathematics 27(1-2), 59–68 (1990)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Cheriyan, J., Vempala, S., Vetta, A.: An approximation algorithm for the minimum-cost k-vertex connected subgraph. SIAM Journal on Computing 32(4), 1050–1055 (2003)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Escoffier, B., Gourvès, L., Monnot, J.: Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs. Journal of Discrete Algorithms 8(1), 36–49 (2010)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Feige, U., Singh, M.: Improved approximation ratios for traveling salesperson tours and paths in directed graphs. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) APPROX and RANDOM 2007. LNCS, vol. 4627, pp. 104–118. Springer, Heidelberg (2007)Google Scholar
  10. 10.
    Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximation for finding a maximum weight Hamiltonian cycle. Operations Research 27(4), 799–809 (1979)CrossRefMATHMathSciNetGoogle 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 (1979)Google Scholar
  12. 12.
    Gimadi, E.K., Serdyukov, A.I.: A problem of finding the maximal spanning connected subgraph with given vertex degrees. In: Operations Reserach Proceedings 2000, pp. 55–59. Springer (2001)Google Scholar
  13. 13.
    Guha, S., Khuller, S.: Improved methods for approximating node weighted steiner trees and connected dominating sets. Information and Computation 150(1), 57–74 (1999)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Kaplan, H., Lewenstein, M., Shafrir, N., Sviridenko, M.I.: Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs. Journal of the ACM 52(4), 602–626 (2005)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Khuller, S., Raghavachari, B.: Improved approximation algorithms for uniform connectivity problems. Journal of Algorithms 21(2), 434–450 (1996)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Khuller, S., Raghavachari, B.: Graph connectivity. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms. Springer (2008)Google Scholar
  17. 17.
    Khuller, S., Vishkin, U.: Biconnectivity approximations and graph carvings. Journal of the ACM 41(2), 214–235 (1994)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Lovász, L., Plummer, M.D.: Matching Theory. North-Holland Mathematics Studies, vol. 121. Elsevier (1986)Google Scholar
  19. 19.
    Marx, D., O’sullivan, B., Razgon, I.: Finding small separators in linear time via treewidth reduction. ACM Transactions on Algorithms 9(4), 30:1–30:35 (2013)Google Scholar
  20. 20.
    Paluch, K., Mucha, M., Mądry, A.: A 7/9 approximation algorithm for the maximum traveling salesman problem. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX and RANDOM 2009. LNCS, vol. 5687, pp. 298–311. Springer, Heidelberg (2009)Google Scholar
  21. 21.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43(3), 425–440 (1991)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Papadimitriou, C.H., Yannakakis, M.: The traveling salesman problem with distances one and two. Mathematics of Operations Research 18(1), 1–11 (1993)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Singh, M., Lau, L.C.: Approximating minimum bounded degree spanning trees to within one of optimal. In: Proc. of the 39th Ann. Int. Symp. on Theory of Computing (STOC), pp. 661–670. ACM (2007)Google Scholar
  24. 24.
    Tutte, W.T.: A short proof of the factor theorem for finite graphs. Canadian Journal of Mathematics 6, 347–352 (1954), http://dx.doi.org/10.4153/CJM-1954-033-3 CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    West, D.B.: Introduction to Graph Theory. Prentice-Hall (2001)Google Scholar
  26. 26.
    Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Kamiel Cornelissen
    • 1
  • Ruben Hoeksma
    • 1
  • Bodo Manthey
    • 1
  • N. S. Narayanaswamy
    • 2
  • C. S. Rahul
    • 2
  1. 1.University of TwenteEnschedeThe Netherlands
  2. 2.Indian Institute of Technology MadrasChennaiIndia

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