Abstract
The Partial Vertex Cover problem is to decide whether a graph contains at most k nodes covering at least t edges. We present deterministic and randomized algorithms with run times of O *(1.396t) and O *(1.2993t), respectively. For graphs of maximum degree three, we show how to solve this problem in O *(1.26t) steps. Finally, we give an O *(3t) algorithm for Exact Partial Vertex Cover, which asks for at most k nodes covering exactly t edges.
Supported by the DFG under grant RO 927/7-1.
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
Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)
Bar-Yehuda, R.: Using homogenous weights for approximating the partial cover problem. In: Proc. of 10th SODA, pp. 71–75 (1999)
Bläser, M.: Computing small partial coverings. Inf. Proc. Letters 85, 327–331 (2003)
Bshouty, N.H., Burroughs, L.: Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 298–308. Springer, Heidelberg (1998)
Cai, L., Chan, S.M., Chan, S.O.: Random separation: A new method for solving fixed-cardinality optimization problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 239–250. Springer, Heidelberg (2006)
Chen, J., Kanj, I.A., Xia, G.: Simplicity is beauty: Improved upper bounds for vertex cover. Technical Report TR05-008, School of CTI, DePaul University (2005)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. Congressus Numerantium 87, 873–921 (1992)
Fomin, F., Grandoni, F., Kratsch, D.: A measure & conquer approach for the analysis of exact algorithms. Technical Report 359, Department of Informatics, University of Bergen (July 2007)
Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. Journal of Algorithms 53, 55–84 (2004)
Guo, J., Niedermeier, R., Wernicke, S.: Parameterized complexity of generalized vertex cover problems. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 36–48. Springer, Heidelberg (2005)
Halperin, E., Srinivasan, R.: Improved approximation algorithms for the partial vertex cover problem. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 185–199. Springer, Heidelberg (2002)
Hochbaum, D.S.: The t-vertex cover problem: Extending the half integrality framework with budget constraints. In: Jansen, K., Rolim, J.D.P. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 111–122. Springer, Heidelberg (1998)
Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)
Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Intuitive algorithms and t-vertex cover. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 598–607. Springer, Heidelberg (2006)
Kneis, J., Mölle, D., Rossmanith, P.: Partial vs. complete domination: t-dominating set. In: van Leeuwen, J., Italiano, G.F., van der Hoek, W., Meinel, C., Sack, H., Plášil, F. (eds.) SOFSEM 2007. LNCS, vol. 4362, pp. 367–376. Springer, Heidelberg (2007)
Monien, B., Speckenmeyer, E.: Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Informatica 22, 115–123 (1985)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
Robson, J.M.: Finding a maximum independent set in time O(2n/4). Technical Report 1251-01, Université Bordeaux I, LaBRI (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kneis, J., Langer, A., Rossmanith, P. (2008). Improved Upper Bounds for Partial Vertex Cover. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2008. Lecture Notes in Computer Science, vol 5344. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92248-3_22
Download citation
DOI: https://doi.org/10.1007/978-3-540-92248-3_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92247-6
Online ISBN: 978-3-540-92248-3
eBook Packages: Computer ScienceComputer Science (R0)