Abstract
The theory of Graph Minors by Robertson and Seymour is one of the deepest and significant theories in modern Combinatorics. This theory has also a strong impact on the recent development of Algorithms, and several areas, like Parameterized Complexity, have roots in Graph Minors. Until very recently it was a common belief that Graph Minors Theory is mainly of theoretical importance. However, it appears that many deep results from Robertson and Seymour’s theory can be also used in the design of practical algorithms. Minor containment testing is one of algorithmically most important and technical parts of the theory, and minor containment in graphs of bounded branchwidth is a basic ingredient of this algorithm. In order to implement minor containment testing on graphs of bounded branchwidth, Hicks [NETWORKS 04] described an algorithm, that in time \(\mathcal{O}(3^{k^2}\cdot (h+k-1)!\cdot m)\) decides if a graph G with m edges and branchwidth k, contains a fixed graph H on h vertices as a minor. That algorithm follows the ideas introduced by Robertson and Seymour in [J’CTSB 95]. In this work we improve the dependence on k of Hicks’ result by showing that checking if H is a minor of G can be done in time \(\mathcal{O}(2^{(2k +1 )\cdot \log k} \cdot h^{2k} \cdot 2^{2h^2} \cdot m)\). Our approach is based on a combinatorial object called rooted packing, which captures the properties of the potential models of subgraphs of H that we seek in our dynamic programming algorithm. This formulation with rooted packings allows us to speed up the algorithm when G is embedded in a fixed surface, obtaining the first single-exponential algorithm for minor containment testing. Namely, it runs in time \(2^{\mathcal{O}(k)} \cdot h^{2k} \cdot 2^{\mathcal{O}(h)} \cdot n\), with n = |V(G)|. Finally, we show that slight modifications of our algorithm permit to solve some related problems within the same time bounds, like induced minor or contraction minor containment.
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
Adler, I., Dorn, F., Fomin, F.V., Sau, I., Thilikos, D.M.: Faster Parameterized Algorithms for Minor Containment (2010), http://users.uoa.gr/~sedthilk/papers/minorch.pdf
Adler, I., Grohe, M., Kreutzer, S.: Computing excluded minors. In: Proc. of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 641–650 (2008)
Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: Handbook of Theoretical Computer Science. Formal Models and Semantics (B), vol. B, pp. 193–242 (1990)
Dawar, A., Grohe, M., Kreutzer, S.: Locally Excluding a Minor. In: Proc. of the 22nd IEEE Symposium on Logic in Computer Science (LICS), pp. 270–279 (2007)
Demaine, E.D., Hajiaghayi, M.T., Kawarabayashi, K.-i.: Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring. In: Proc. of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 637–646 (2005)
Diestel, R.: Graph Theory, vol. 173. Springer, Heidelberg (2005)
Dorn, F.: Planar Subgraph Isomorphism Revisited. In: Proc. of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS), pp. 263–274 (2010)
Fellows, M.R., Langston, M.A.: Nonconstructive tools for proving polynomial-time decidability. Journal of the ACM 35(3), 727–739 (1988)
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2008)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)
Hicks, I.V.: Branch decompositions and minor containment. Networks 43(1), 1–9 (2004)
Kawarabayashi, K.-i., Reed, B.A.: Hadwiger’s conjecture is decidable. In: Proc. of the 41st Annual ACM Symposium on Theory of Computing (STOC), pp. 445–454 (2009)
Matoušek, J., Thomas, R.: On the complexity of finding iso- and other morphisms for partial k-trees. Discrete Mathematics 108, 143–364 (1992)
Robertson, N., Seymour, P.: Graph Minors. XIII. The Disjoint Paths Problem. Journal of Combinatorial Theory, Series B 63(1), 65–110 (1995)
Robertson, N., Seymour, P.D.: Graph Minors. XX. Wagner’s conjecture. J. Comb. Theory, Ser. B 92(2), 325–357 (2004)
Rué, J., Sau, I., Thilikos, D.M.: Dynamic Programming for Graphs on Surfaces. To appear in Proc. of the 37th International Colloquium on Automata, Languages and Programming, ICALP (2010)
Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Adler, I., Dorn, F., Fomin, F.V., Sau, I., Thilikos, D.M. (2010). Faster Parameterized Algorithms for Minor Containment. In: Kaplan, H. (eds) Algorithm Theory - SWAT 2010. SWAT 2010. Lecture Notes in Computer Science, vol 6139. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13731-0_31
Download citation
DOI: https://doi.org/10.1007/978-3-642-13731-0_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13730-3
Online ISBN: 978-3-642-13731-0
eBook Packages: Computer ScienceComputer Science (R0)