Abstract
Given a graph G=(V,E) and a tree T=(V,F) with E∩F=∅ such that G + T = (V,F∪E) is 2-edge-connected, we consider the problem of finding a smallest 2-edge-connected spanning subgraph (V,F∪\( E' \)) of G + T containing T. The problem, which is known to be NP-hard, admits a 2-approximation algorithm. However, obtaining a factor better than 2 for this problem has been one of the main open problems in the graph augmentation problem. In this paper, we present an O \( \left( {\sqrt n m} \right) \) time \( \frac{{12}} {7} \)-approximation algorithm for this problem, where \( n = \left| V \right| \) and \( m = \left| {E \cup F} \right| \)
This research was partially supported by the Scientific Grant-in-Aid from Ministry of Education, Science, Sports and Culture of Japan, and the subsidy from the Inamori Foundation.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
J. Cheriyan, T. Jordán and R. Ravi: “On 2-coverings and 2-packings in laminar families,” 7th Annual European Symposium on Algorithms, July 16—18, 1999, Prague, Czech Republic (1999) (to appear).
J. Cheriyan, A. Sebö and Z. Szigeti: “An improved approximation algorithm for minimum size 2-edge connected spanning subgraphs,” Lecture Notes in Computer Science, 1412, Springer-Verlag, IPCO'98 (1998) 126–136.
J. Cheriyan and R. Thurimella: “Approximating minimum-size k-connected spanning subgraphs via matching,” Proc. 37th IEEE Symp. on Found. Comp. Sci. (1996) 292–301.
E. A. Dinits, A. V. Karzanov and M. V. Lomonosov: “On the structure of a family of minimal weighted cuts in a graph,” Studies in Discrete Optimization (in Russian) A.A. Fridman (Ed.) Nauka, Moscow (1976) 290–306.
K. P. Eswaran and R. E. Tarjan: “Augmentation problems,” SIAM J. Computing 5 (1976) 653–665.
G. N. Frederickson and J. JáJá: “On the relationship between the biconnectivity augmentation problems,” SIAM J. Computing 10 (1981) 270–283.
H. N. Gabow: “Applications of a poset representation to edge connectivity and graph rigidity,” Proc. 32nd IEEE Symp. on Found. Comp. Sci. (1991) 812–821.
S. Khuller: “Approximation algorithms for finding highly connected subgraphs,” in Approximation Algorithms, D. Hochbaum (Eds.), PWS publishing company (1997) 236–265.
S. Khuller and R. Thurimella: “Approximation algorithms for graph augmentation,” Proc. 19th International Colloquium on Automata, Languages and Programming Conference (1992) 330–341.
S. Khuller and U. Vishkin: “Biconnectivity approximations and graph carvings,” J. ACM, 41 (1994) 214–235.
S. Micali and V. V. Vazirani: “An \( O\left( {\sqrt {\left| V \right|} \left| E \right|} \right) \) algorithm for finding maximum matching in general graph,” Proc. 21st IEEE Symp. on Found. Comp. Sci. (1980) 17–27.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nagamochi, H., Ibaraki, T. (1999). An Approximation for Finding a Smallest 2-Edge-Connected Subgraph Containing a Specified Spanning Tree. In: Asano, T., Imai, H., Lee, D.T., Nakano, Si., Tokuyama, T. (eds) Computing and Combinatorics. COCOON 1999. Lecture Notes in Computer Science, vol 1627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48686-0_3
Download citation
DOI: https://doi.org/10.1007/3-540-48686-0_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66200-6
Online ISBN: 978-3-540-48686-2
eBook Packages: Springer Book Archive