Approximation Schemes for Minimum 2-Connected Spanning Subgraphs in Weighted Planar Graphs

We present new approximation schemes for various classical problems of finding the minimum-weight spanning subgraph in edge-weighted undirected planar graphs that are resistant to edge or vertex removal. 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, which augments any connected subgraph A of a weighted planar graph G to a (1+e)-spanner of G with total weight bounded by weight(A)/e. From this we derive quasi-polynomial time approximation schemes for the problems of finding the minimum-weight 2-edge-connected or biconnected spanning subgraph in planar graphs. We also design approximation schemes for the minimum-weight 1-2-connectivity problem, which is the variant of the survivable network design problem where vertices have 1 or 2 connectivity constraints. Prior to our work, for all these problems no polynomial or quasi-polynomial time algorithms were known to achieve an approximation ratio better than 2.

[1]  Hairong Zhao,et al.  A Well-Connected Separator for Planar Graphs , 2004 .

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

[3]  M. Stoer.  Design of Survivable Networks , 1993 .

[4]  David P. Williamson,et al.  An iterative rounding 2-approximation algorithm for the element connectivity problem , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

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

[6]  Pawel Winter,et al.  Steiner problem in networks: A survey , 1987, Networks.

[7]  Joachim Gudmundsson,et al.  Improved Greedy Algorithms for Constructing Sparse Geometric Spanners , 2000, SWAT.

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

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

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

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

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

[13]  Piotr Indyk,et al.  Approximate nearest neighbors: towards removing the curse of dimensionality , 1998, STOC '98.

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

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

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

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

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

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

[20]  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).

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

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

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

[24]  Joseph S. B. Mitchell,et al.  Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems , 1999, SIAM J. Comput..

[25]  H. Pollak,et al.  Steiner Minimal Trees , 1968 .

[26]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

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

[28]  Martin Grötschel,et al.  Polyhedral and Computational Investigations for Designing Communication Networks with High Survivability Requirements , 1995, Oper. Res..

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

[30]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[31]  David P. Williamson,et al.  A primal-dual approximation algorithm for generalized steiner network problems , 1993, Comb..

[32]  Joseph JáJá,et al.  On the Relationship between the Biconnectivity Augmentation and Traveling Salesman Problems , 1982, Theor. Comput. Sci..

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

[34]  Lisa Fleischer.  A 2-Approximation for Minimum Cost {0, 1, 2} Vertex Connectivity , 2001, IPCO.

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

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

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

[38]  Klaus Jansen,et al.  Approximation Algorithms for Combinatorial Optimization , 2000 .

[39]  Dana S. Richards,et al.  Steiner tree problems , 1992, Networks.

[40]  Hans Jürgen Prömel,et al.  The Steiner Tree Problem , 2002 .

[41]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

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

[43]  Sunil Chopra,et al.  A branch-and-cut approach for minimum cost multi-level network design , 2002, Discret. Math..

[44]  H. Prömel,et al.  The Steiner Tree Problem: A Tour through Graphs, Algorithms, and Complexity , 2002 .