Approximate Minimum 2-Connected Subgraphs in Weighted Planar Graphs

We consider the problems of finding the minimum-weight 2-connected spanning subgraph in edge-weighted planar graphs and its variations. We first give a PTAS for the problem of finding minimum-weight 2-edge-connected spanning subgraphs where duplicate edges are allowed. Then we present a new greedy spanner construction for edge-weighted planar graphs. From this we derive quasi-polynomial time approximation schemes for the problems of finding the lightest 2-edge-connected or biconnected spanning subgraph in such graphs. We also design efficient approximation schemes for the variant where vertices have non-uniform (1 or 2) connectivity constraints.

[1]  Harold N. Gabow.  Better performance bounds for finding the smallest k-edge connected spanning subgraph of a multigraph , 2003, SODA '03.

[2]  David P. Williamson,et al.  An efficient approximation algorithm for the survivable network design problem , 1998, Math. Program..

[3]  Philip N. Klein,et al.  A polynomial-time approximation scheme for weighted planar graph TSP , 1998, SODA '98.

[4]  Harold N. Gabow,et al.  An ear decomposition approach to approximating the smallest 3-edge connected spanning subgraph of a multigraph , 2002, SODA '02.

[5]  Andrzej Lingas,et al.  Fast Approximation Schemes for Euclidean Multi-connectivity Problems , 2000, ICALP.

[6]  Michal Penn,et al.  Improved Approximation Algorithms for Weighted 2- and 3-Vertex Connectivity Augmentation Problems , 1997, J. Algorithms.

[7]  Guy Kortsarz,et al.  Approximation algorithm for k-node connected subgraphs via critical graphs , 2004, STOC '04.

[8]  C. Monma,et al.  Methods for Designing Communications Networks with Certain Two-Connected Survivability Constraints , 1989, Oper. Res..

[9]  S. Khuller.  Approximation algorithms for finding highly connected subgraphs , 1996 .

[10]  G. F. J-SJ,et al.  A 2-approximation for Minimum Cost {0, 1, 2} Vertex Connectivity , 2001 .

[11]  Kamal Jain,et al.  A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[12]  Giri Narasimhan,et al.  Efficient algorithms for constructing fault-tolerant geometric spanners , 1998, STOC '98.

[13]  Panos M. Pardalos,et al.  Design of survivable networks , 2006 .

[14]  David P. Dobkin,et al.  On sparse spanners of weighted graphs , 1993, Discret. Comput. Geom..

[15]  Santosh S. Vempala,et al.  An Approximation Algorithm for the Minimum-Cost k-Vertex Connected Subgraph , 2003, SIAM J. Comput..

[16]  Andrzej Lingas,et al.  Polynomial-Time Approximation Schemes for the Euclidean Survivable Network Design Problem , 2002, ICALP.

[17]  Santosh S. Vempala,et al.  Factor 4/3 approximations for minimum 2-connected subgraphs , 2000, APPROX.

[18]  Andrzej Lingas,et al.  On approximability of the minimum-cost k-connected spanning subgraph problem , 1999, SODA '99.

[19]  Martin Grötschel,et al.  Integer Polyhedra Arising from Certain Network Design Problems with Connectivity Constraints , 1990, SIAM J. Discret. Math..

[20]  Balaji Raghavachari,et al.  A 5/4-approximation algorithm for minimum 2-edge-connectivity , 2003, SODA '03.

[21]  Joseph Cheriyan,et al.  Approximating Minimum-Size k-Connected Spanning Subgraphs via Matching , 1998, Electron. Colloquium Comput. Complex..

[22]  Satish Rao,et al.  Approximating geometrical graphs via “spanners” and “banyans” , 1998, STOC '98.

[23]  Artur Czumaj,et al.  Approximation schemes for minimum 2-edge-connected and biconnected subgraphs in planar graphs , 2004, SODA '04.

[24]  Samir Khuller,et al.  Biconnectivity approximations and graph carvings , 1992, STOC '92.

[25]  Martin Grötschel,et al.  Computational Results with a Cutting Plane Algorithm for Designing Communication Networks with Low-Connectivity Constraints , 1992, Oper. Res..

[26]  Piotr Krysta.  Approximating minimum size {1, 2}-connected network , 2003, Discret. Appl. Math..