Abstract
The dichotomy conjecture for the parameterized embedding problem states that the problem of deciding whether a given graph G from some class \(\mathcal K\) of “pattern graphs” can be embedded into a given graph H (that is, is isomorphic to a subgraph of H) is fixed-parameter tractable if \(\mathcal K\) is a class of graphs of bounded tree width and \(W [1]\)-complete otherwise.
Towards this conjecture, we prove that the embedding problem is \(W [1]\)-complete if \(\mathcal K\) is the class of all grids or the class of all walls.
Full version available at https://arxiv.org/abs/1703.06423.
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 subscriptionsNotes
- 1.
There is a minor issue here regarding the computability of the class \(\mathcal K\): if we want to include classes \(\mathcal K\) that are not recursively enumerable here then we need the nonuniform notion of fixed-parameter tractability [11].
- 2.
If \(\mathcal K\) is not recursively enumerable, there is still a “non-uniform” hardness result. See [13] for a discussion.
References
Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)
Bodirsky, M., Grohe, M.: Non-dichotomies in constraint satisfaction complexity. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5126, pp. 184–196. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70583-3_16
Chen, H., Müller, M.: One hierarchy spawns another: graph deconstructions and the complexity classification of conjunctive queries. In: Proceedings of the Joint Meeting of the 23rd EACSL Annual Conference on Computer Science Logic and the 29th Annual ACM/IEEE Symposium on Logic in Computer Science, pp. 32:1–32:10 (2014)
Chen, Y., Thurley, M., Weyer, M.: Understanding the complexity of induced subgraph isomorphisms. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5125, pp. 587–596. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70575-8_48
Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3
Dalmau, V., Kolaitis, P.G., Vardi, M.Y.: Constraint satisfaction, bounded treewidth, and finite-variable logics. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 310–326. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-46135-3_21
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-662-53622-3
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999). https://doi.org/10.1007/978-1-4612-0515-9
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: on completeness for W[1]. Theoret. Comput. Sci. 141, 109–131 (1995)
Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl. 3, 1–27 (1999)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-29953-X
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Grohe, M.: The complexity of homomorphism, constraint satisfaction problems seen from the other side. J. ACM 54(1), 1:1–1:24 (2007)
Grohe, M., Schwentick, T., Segoufin, L.: When is the evaluation of conjunctive queries tractable. In: Proceedings on the 33rd Annual ACM Symposium on Theory of Computing, Heraklion, Crete, Greece, 6–8 July 2001, pp. 657–666 (2001)
Jansen, B.M.P., Marx, D.: Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, 4–6 January 2015, pp. 616–629 (2015)
Lin, B.: The parameterized complexity of k-biclique. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, 4–6 January 2015, pp. 605–615 (2015)
Marx, D., Pilipczuk, M.: Everything you always wanted to know about the parameterized complexity of subgraph isomorphism (but were afraid to ask). ArXiv (CoRR), abs/1307.2187 (2013)
Matula, D.W.: Subtree isomorphism in \(o(n^{5/2})\). In: Alspach, P.H.B., Miller, D. (eds.) Algorithmic Aspects of Combinatorics. Annals of Discrete Mathematics, vol. 2, pp. 91–106. Elsevier (1978)
Monien, B.: How to find longest paths efficiently. In: Analysis and Design of Algorithms for Combinatorial Problems. North Holland Mathematics Studies, vol. 109, pp. 239–254. North Holland (1985)
Plehn, J., Voigt, B.: Finding minimally weighted subgraphs. In: Möhring, R.H. (ed.) WG 1990. LNCS, vol. 484, pp. 18–29. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-53832-1_28
Robertson, N., Seymour, P.D.: Graph minors V. Excluding a planar graph. J. Comb. Theory Ser. B 41, 92–114 (1986)
Ullman, J.R.: An algorithm for subgraph isomorphism. J. ACM 23(1), 31–42 (1976)
Acknowledgement
Yijia Chen is partially supported by the Sino-German Center for Research Promotion (CDZ 996) and National Nature Science Foundation of China (Project 61373029). Bingkai Lin is partially supported by the JSPS KAKENHI Grant (JP16H07409) and JST ERATO Grant (JPMJER1201) of Japan.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Chen, Y., Grohe, M., Lin, B. (2017). The Hardness of Embedding Grids and Walls. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-68705-6_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68704-9
Online ISBN: 978-3-319-68705-6
eBook Packages: Computer ScienceComputer Science (R0)