Abstract
We provide parameterized algorithms for nonblocker, the parametric dual of the well known dominating set problem. We exemplify three methodologies for deriving parameterized algorithms that can be used in other circumstances as well, including the (i) use of extremal combinatorics (known results from graph theory) in order to obtain very small kernels, (ii) use of known exact algorithms for the (nonparameterized) minimum dominating set problem, and (iii) use of exponential space. Parameterized by the size k d of the non-blocking set, we obtain an algorithm that runs in time \({\mathcal O}^{*}(1.4123^{k_{d}})\) when allowing exponential space.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed Parameter Algorithms for Dominating Set and Related Problems on Planar Graphs. Algorithmica 33, 461–493 (2002)
Ausiello, G., Creczenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation; Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (1999)
Blank, M.: An Estimate of the External Stability Number of a Graph without Suspended Vertices (in Russian). Prikl. Math. i Programmirovanie Vyp 10, 3–11 (1973)
Chen, J., Fernau, H., Kanj, I.A., Xia, G.: Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 269–280. Springer, Heidelberg (2005)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Fernau, H.: Roman Domination: a Parameterized Perspective. In: Wiedermann, J., et al. (eds.) SOFSEM 2006. LNCS, vol. 3831. Springer, Heidelberg (2006)
Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and Conquer: Domination – a Case Study. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 191–203. Springer, Heidelberg (2005)
Fomin, F.V., Grandoni, F., Kratsch, D.: Some New Techniques in Design and Analysis of Exact (Exponential) Algorithms. Technical Report 307, Department of Informatics, University of Bergen (2005)
Fomin, F.V., Thilikos, D.: A Simple and Fast Approach for Solving Problems on Planar Graphs. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 56–67. Springer, Heidelberg (2004)
Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, New York (1979)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, New York (1998)
Hedetniemi, S.T.: A Max-Min Relationship between Matchings and Domination in Graphs. Congressus Numerantium 40, 23–34 (1983)
Kikuno, T., Yoshida, N., Kakuda, Y.: The NP-Completeness of the Dominating Set Problem in Cubic Planar Graphs. Transactions of the Institute of Electronics and Communication Engineers of Japan E63(6), 443–444 (1980)
Manlove, D.F.: On the Algorithmic Complexity of Twelve Covering and Independence Parameters of Graphs. Discrete Applied Mathematics 91, 155–175 (1999)
McCuaig, B., Shepherd, B.: Domination in Graphs of Minimum Degree Two. Journal of Graph Theory 13, 749–762 (1989)
Nieminen, J.: Two Bounds for the Domination Number of a Graph. Journal of the Institute of Mathematics and its Applications 14, 183–187 (1974)
Ore, O.: Theory of Graphs, Colloquium Publications. American Mathematical Society XXXVIII (1962)
Prieto, E.: Systematic Kernelization in FPT Algorithm Design. PhD thesis, The University of Newcastle, Australia (2005)
Reed, B.: Paths, Stars, and the Number Three. Combinatorics, Probability and Computing 5, 277–295 (1996)
Thomassé, S., Yeo, Y.: Total Domination of Graphs and Small Transversals of Hypergraphs. To appear in Combinatorica (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dehne, F., Fellows, M., Fernau, H., Prieto, E., Rosamond, F. (2006). nonblocker: Parameterized Algorithmics for minimum dominating set . In: Wiedermann, J., Tel, G., Pokorný, J., Bieliková, M., Štuller, J. (eds) SOFSEM 2006: Theory and Practice of Computer Science. SOFSEM 2006. Lecture Notes in Computer Science, vol 3831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11611257_21
Download citation
DOI: https://doi.org/10.1007/11611257_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31198-0
Online ISBN: 978-3-540-32217-7
eBook Packages: Computer ScienceComputer Science (R0)