Fixed-Parameter Algorithms for Minimum-Cost Edge-Connectivity Augmentation

We consider connectivity-augmentation problems in a setting where each potential new edge has a non-negative cost associated with it, and the task is to achieve a certain connectivity target with at most p new edges of minimum total cost. The main result is that the minimum cost augmentation of edge-connectivity from k − 1 to k with at most p new edges is fixed-parameter tractable parameterized by p and admits a polynomial kernel. We also prove the fixed-parameter tractability of increasing edge connectivity from 0 to 2 and increasing node connectivity from 1 to 2.

[1]  László A. Végh,et al.  Augmenting undirected node-connectivity by one , 2010, STOC '10.

[2]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[3]  Jaroslav Nesetril,et al.  Otakar Boruvka on minimum spanning tree problem Translation of both the 1926 papers, comments, history , 2001, Discret. Math..

[4]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[5]  András Frank,et al.  Minimal Edge-Coverings of Pairs of Sets , 1995, J. Comb. Theory B.

[6]  R. Ravi,et al.  When trees collide: an approximation algorithm for the generalized Steiner problem on networks , 1991, STOC '91.

[7]  Jiong Guo,et al.  Kernelization and complexity results for connectivity augmentation problems , 2010, Networks.

[8]  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).

[9]  András Frank.  Augmenting Graphs to Meet Edge-Connectivity Requirements , 1992, SIAM J. Discret. Math..

[10]  Saket Saurabh,et al.  Lower bounds on kernelization , 2011, Discret. Optim..

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

[12]  Guy Kortsarz,et al.  Approximating node connectivity problems via set covers , 2000, APPROX.

[13]  Guy Kortsarz,et al.  Approximating Minimum-Cost Connectivity Problems , 2007, Handbook of Approximation Algorithms and Metaheuristics.

[14]  Petr A. Golovach,et al.  Parameterized Algorithms to Preserve Connectivity , 2014, ICALP.

[15]  László A. Végh,et al.  Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs , 2014, SIAM J. Comput..

[16]  R. Ravi,et al.  When Trees Collide: An Approximation Algorithm for the Generalized Steiner Problem on Networks , 1995, SIAM J. Comput..

[17]  David P. Williamson,et al.  A general approximation technique for constrained forest problems , 1992, SODA '92.

[18]  Akira Nakamura,et al.  Edge-Connectivity Augmentation Problems , 1987, J. Comput. Syst. Sci..

[19]  Tibor Jordán,et al.  On the Optimal Vertex-Connectivity Augmentation , 1995, J. Comb. Theory B.

[20]  Guy Kortsarz,et al.  Approximating Node Connectivity Problems via Set Covers , 2003, Algorithmica.

[21]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[22]  Bill Jackson,et al.  Independence free graphs and vertex connectivity augmentation , 2001, J. Comb. Theory, Ser. B.

[23]  T. Hsu,et al.  On four-connecting a triconnected graph , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[24]  A. Frank.  Connections in Combinatorial Optimization , 2011 .

[25]  András Frank,et al.  An application of simultaneous diophantine approximation in combinatorial optimization , 1987, Comb..

[26]  Saket Saurabh,et al.  Kernelization - Preprocessing with a Guarantee , 2012, The Multivariate Algorithmic Revolution and Beyond.

[27]  Hiroshi Nagamochi,et al.  An approximation for finding a smallest 2-edge-connected subgraph containing a specified spanning tree , 1999, Discret. Appl. Math..

[28]  László A. Végh,et al.  Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs , 2012, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.