Abstract
We establish refined search tree techniques for the parameterized DOMINATING SET problem on planar graphs. We derive a fixed parameter algorithm with running time O(8 k n), where k is the size of the dominating set and n is the number of vertices in the graph. For our search tree, we firstly provide a set of reduction rules. Secondly, we prove an intricate branching theorem based on the Euler formula. In addition, we give an example graph showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final algorithm is very easy (to implement); its analysis, however, is involved.
Supported by the Deutsche Forschungsgemeinschaft (DFG), research project PEAL (Parameterized complexity and Exact ALgorithms), NI 369/1-1.
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J. Alber, H. L. Bodlaender, H. Fernau, and R. Niedermeier. Fixed parameter algorithms for planar dominating set and related problems. In 7th Scandinavian Workshop on Algorithm Theory SWAT, volume 1851 of LNCS, pages 97–110, Springer-Verlag, 2000. A lng version has been accepted for publication in Algorithmica.
J. Alber, H. Fernau, and R. Niedermeier. Graph separators: a parameterized view. Technical Report WSI-2001-8, Universität Tübingen (Fed. Rep. of Germany), Wilhelm-Schickard-Institut für Informatik, 2001. Extended abstract accepted at COCOON 2001, to appear in LNCS, Springer-Verlag, August 2001.
J. Alber, H. Fernau, and R. Niedermeier. Parameterized complexity: exponential speedup for planar graph problems. Technical Report TR01-023, ECCC Reports, Trier (Fed. Rep. of Germany), March 2001. Extended abstract accepted at ICALP 2001, to appear in LNCS, Springer-Verlag, July 2001.
L. Cai, M. Fellows, D. Juedes, and F. Rosamond. Efficient polynomial-time approximation schemes for problems on planar graph structures: upper and lower bounds. Manuscript, May 2001.
L. Cai and D. Juedes. Subexponential parameterized algorithms collapse the W-hierarchy. Extended abstract accepted at ICALP 2001, to appear in LNCS, Springer-Verlag, July 2001.
R. Diestel. Graph Theory. Springer-Verlag, 1997.
R. G. Downey and M. R. Fellows. Parameterized computational feasibility. In Feasible Mathematics II, pages 219–244. Birkhäuser, 1995.
R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer-Verlag, 1999.
R. G. Downey, M. R. Fellows, and U. Stege. Parameterized complexity: A framework for systematically confronting computational intractability. DIM ACS Series in Discrete Mathematics and Theoretical Computer Science, 49:49–99, 1999.
R. Niedermeier and P. Rossmanith. A general method to speed up fixed-parameter-tractable algorithms. Information Processing Letters, 73:125–129, 2000.
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Alber, J. et al. (2001). Refined Search Tree Technique for Dominating Set on Planar Graphs. In: Sgall, J., Pultr, A., Kolman, P. (eds) Mathematical Foundations of Computer Science 2001. MFCS 2001. Lecture Notes in Computer Science, vol 2136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44683-4_11
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DOI: https://doi.org/10.1007/3-540-44683-4_11
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