Complexity and Approximation Results for the Connected Vertex Cover Problem

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs).

[1]  Max Planck,et al.  Complexity of approximating bounded variants of optimization problems , 2008 .

[2]  Teofilo F. Gonzalez,et al.  Complexity of the minimum-length corridor problem , 2007, Comput. Geom..

[3]  Hans L. Bodlaender,et al.  A Partial k-Arboretum of Graphs with Bounded Treewidth , 1998, Theor. Comput. Sci..

[4]  Gerhard J. Woeginger,et al.  Local search for the minimum label spanning tree problem with bounded color classes , 2003, Oper. Res. Lett..

[5]  Niklaus Wirth,et al.  Algorithms and Data Structures , 1989, Lecture Notes in Computer Science.

[6]  K. Onaga,et al.  Vertex covers and connected vertex covers in 3-connected graphs , 1991, 1991., IEEE International Sympoisum on Circuits and Systems.

[7]  Venkatesan Guruswami,et al.  A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover , 2005, SIAM J. Comput..

[8]  Giorgio Gambosi,et al.  Complexity and Approximation , 1999, Springer Berlin Heidelberg.

[9]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[10]  A. Brandstädt,et al.  Graph Classes: A Survey , 1987 .

[11]  Giorgio Gambosi,et al.  Complexity and approximation: combinatorial optimization problems and their approximability properties , 1999 .

[12]  Yoji Kajitani,et al.  On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three , 1988, Discret. Math..

[13]  Esther M. Arkin,et al.  Approximating the Tree and Tour Covers of a Graph , 1993, Inf. Process. Lett..

[14]  Toshihiro Fujito.  How to Trim an MST: A 2-Approximation Algorithm for Minimum Cost Tree Cover , 2006, ICALP.

[15]  David S. Johnson,et al.  The Rectilinear Steiner Tree Problem is NP Complete , 1977, SIAM Journal of Applied Mathematics.

[16]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[17]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[18]  Refael Hassin,et al.  Approximation algorithms and hardness results for labeled connectivity problems , 2006, J. Comb. Optim..

[19]  Erik D. Demaine,et al.  Bidimensionality: new connections between FPT algorithms and PTASs , 2005, SODA '05.

[20]  Luca Trevisan,et al.  Non-approximability results for optimization problems on bounded degree instances , 2001, STOC '01.

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

[22]  Stephan Kreutzer,et al.  Approximation Schemes for First-Order Definable Optimisation Problems , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).

[23]  Rolf Niedermeier,et al.  Exact Algorithms for Generalizations of Vertex Cover , 2005 .

[24]  T Hemalatha,et al.  Connected Vertex Cover in 2-Connected Planar Graph with Maximum Degree 4 is NP-complete , 2007 .

[25]  Rolf Niedermeier,et al.  Parameterized Complexity of Generalized Vertex Cover Problems , 2005, WADS.

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

[27]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[28]  Carla D. Savage,et al.  Depth-First Search and the Vertex Cover Problem , 1982, Inf. Process. Lett..

[29]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[30]  Fanica Gavril,et al.  Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph , 1972, SIAM J. Comput..

[31]  Miroslav Chlebík,et al.  Approximation Hardness of Dominating Set Problems , 2004, ESA.

[32]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[33]  Stefan Richter,et al.  Enumerate and Expand: Improved Algorithms for Connected Vertex Cover and Tree Cover , 2006, Theory of Computing Systems.

[34]  Viggo Kann,et al.  Some APX-completeness results for cubic graphs , 2000, Theor. Comput. Sci..

[35]  Reuven Bar-Yehuda,et al.  On approximating a vertex cover for planar graphs , 1982, STOC '82.

[36]  Ojas Parekh,et al.  Improved Approximations for Tour and Tree Covers , 2000, Algorithmica.

[37]  Stefan Richter,et al.  Enumerate and Expand: New Runtime Bounds for Vertex Cover Variants , 2006, COCOON.

[38]  David Manlove,et al.  Vertex and Edge Covers with Clustering Properties: Complexity and Algorithms , 2009, ACiD.

[39]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..