Abstract
Recently, various linear problem kernels for NP-hard planar graph problems have been achieved, finally resulting in a meta-theorem for classification of problems admitting linear kernels. Almost all of these results are based on a so-called region decomposition technique. In this paper, we introduce a simple partition of the vertex set to analyze kernels for planar graph problems which admit the distance property with small constants. Without introducing new reduction rules, this vertex partition directly leads to improved kernel sizes for several problems. Moreover, we derive new kernelization algorithms for Connected Vertex Cover, Edge Dominating Set, and Maximum Triangle Packing problems, further improving the kernel size upper bounds for these problems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial-time data reduction for dominating set. J. ACM 51(3), 363–384 (2004)
Bodlaender, H.L.: Kernelization: New upper and lower bound techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)
Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) Kernelization. In: FOCS 2009, pp. 629–638. IEEE Computer Society, Los Alamitos (2009)
Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and ids. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 378–389. Springer, Heidelberg (2009)
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)
Gu, Q., Imani, N.: Connectivity is not a limit for kernelization: Planar connected dominating set. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 26–37. Springer, Heidelberg (2010)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(1), 31–45 (2007)
Guo, J., Niedermeier, R.: Linear problem kernels for NP-hard problems on planar graphs. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 375–386. Springer, Heidelberg (2007)
Guo, J., Niedermeier, R., Wernicke, S.: Fixed-parameter tractability results for full-degree spanning tree and its dual. Networks 56(2), 116–130 (2010)
Lokshtanov, D., Mnich, M., Saurabh, S.: Linear kernel for planar connected dominating set. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 281–290. Springer, Heidelberg (2009)
Luo, W., Wang, J., Feng, Q., Guo, J., Chen, J.: An improved kernel for planar connected dominating set. In: Ogihara, M., Tarui, J. (eds.) TAMC 2011. LNCS, vol. 6648, pp. 70–81. Springer, Heidelberg (2011)
Moser, H.: A problem kernelization for graph packing. In: Nielsen, M., Kučera, A., Miltersen, P.B., Palamidessi, C., Tůma, P., Valencia, F. (eds.) SOFSEM 2009. LNCS, vol. 5404, pp. 401–412. Springer, Heidelberg (2009)
Moser, H., Sikdar, S.: The parameterized complexity of the induced matching problem. Discrete Applied Mathematics 157(4), 715–727 (2009)
Prieto, E.: Systematic Kernelization in FPT Algorithm Design. Ph.D. thesis, Department of Computer Science, University of Newcastle, Australia (2005)
Privadarsini, P., Hemalatha, T.: Connected vertex cover in 2-connected planar graph with maximum degree 4 is NP-complete. International Journal of Mathematical, Physical and Engineering Sciences 2(1), 51–54 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag GmbH Berlin Heidelberg
About this paper
Cite this paper
Wang, J., Yang, Y., Guo, J., Chen, J. (2011). Linear Problem Kernels for Planar Graph Problems with Small Distance Property. In: Murlak, F., Sankowski, P. (eds) Mathematical Foundations of Computer Science 2011. MFCS 2011. Lecture Notes in Computer Science, vol 6907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22993-0_53
Download citation
DOI: https://doi.org/10.1007/978-3-642-22993-0_53
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22992-3
Online ISBN: 978-3-642-22993-0
eBook Packages: Computer ScienceComputer Science (R0)