Abstract
Finding a minimum size 2-vertex connected spanning subgraph of a k-vertex connected graph G = (V,E) with n vertices and m edges is known to be NP-hard and APX-hard, as well as approximable in O(n 2 m) time within a factor of 4/3. Interestingly, the problem remains NP-hard even if a Hamiltonian path of G is given as part of the input. For this input-enriched version of the problem, we provide in this paper a linear time and space algorithm which approximates the optimal solution by a factor of no more than min \({\{\frac{5}{4},\frac{2k-1}{2(k-1)}\}}\).
This work has been partially supported by the Research Project GRID.IT, funded by the Italian Ministry of Education, University and Research.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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
Böchenhauer, H.-J., Bongartz, D., Hromkovič, J., Klasing, R., Proietti, G., Seibert, S., Unger, W.: On the hardness of constructing minimal 2-connected spanning subgraphs in complete graphs with sharpened triangle inequality, Theoretical Computer Science, 326, 137–153 (2003); A preliminary version appeared. In: Agrawal, M., Seth, A.K. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 59–70. Springer, Heidelberg (2002)
Czumaj, A., Lingas, A.: On approximability of the minimum-cost k-connected spanning subgraph problem. In: Proc. of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1999), pp. 281–290. ACM Press, New York (1999)
Diestel, R.: Graph theory. electronic edn. Springer, Heidelberg (2000)
Frederickson, G.N., Jájá, J.: On the relationship between the biconnectivity augmentation and traveling salesman problems. Theoretical Computer Science 19, 189–201 (1982)
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)
Garg, N., Santosh, V.S., Singla, A.: Improved approximation algorithms for biconnected subgraphs via better lower bounding techniques. In: Proc. of the 4th ACM-SIAM Symp. on Discrete Algorithms (SODA 1993), pp. 103–111. ACM Press, New York (1993)
Jothi, R., Raghavachari, B., Varadarajan, S.: A 5/4-approximation algorithm for minimum 2-edge-connectivity. In: Proc. of the 14th ACM-SIAM Symp. on Discrete Algorithms (SODA 2003), pp. 725–734. ACM Press, New York (2003)
Jothi, R.: Personal communication (2004)
Khuller, S.: Approximation algorithms for finding highly connected subgraphs. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-Hard Problems, PWS Publishing Company, Boston (1996)
Khuller, S., Vishkin, U.: Biconnectivity approximations and graph carvings. J. ACM 41(2), 214–235 (1994)
Kortsarz, G., Krauthgamer, R., Lee, J.R.: Hardness of approximation for vertex-connectivity network-design problems. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 185–199. Springer, Heidelberg (2002)
Papadimitriou, C.H., Steiglitz, K.: Some complexity results for the Traveling Salesman Problem. In: Proc. of the 8th ACM Symp. on Theory of Computing (STOC 1976), pp. 1–9. ACM Press, New York (1976)
Vempala, S., Vetta, A.: Factor 4/3 approximations for minimum 2-connected subgraphs. In: Jansen, K., Khuller, S. (eds.) APPROX 2000. LNCS, vol. 1913, pp. 262–273. Springer, Heidelberg (2000)
Whitney, H.: Nonseparable and planar graphs. Trans. Amer. Math. Soc. 34, 339–362 (1932)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bilò, D., Proietti, G. (2005). A \(\frac{5}{4}\)-Approximation Algorithm for Biconnecting a Graph with a Given Hamiltonian Path. In: Persiano, G., Solis-Oba, R. (eds) Approximation and Online Algorithms. WAOA 2004. Lecture Notes in Computer Science, vol 3351. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31833-0_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-31833-0_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24574-2
Online ISBN: 978-3-540-31833-0
eBook Packages: Computer ScienceComputer Science (R0)