Understanding the Complexity of Induced Subgraph Isomorphisms
We study left-hand side restrictions of the induced subgraph isomorphism problem: Fixing a class , for given graphs G and arbitrary H we ask for induced subgraphs of H isomorphic to G.
For the homomorphism problem this kind of restriction has been studied by Grohe and Dalmau, Kolaitis and Vardi for the decision problem and by Dalmau and Jonsson for its counting variant.
We give a dichotomy result for both variants of the induced subgraph isomorphism problem. Under some assumption from parameterized complexity theory, these problems are solvable in polynomial time if and only if contains no arbitrarily large graphs.
All classifications are given by means of parameterized complexity. The results are presented for arbitrary structures of bounded arity which implies, for example, analogous results for directed graphs.
Furthermore, we show that no such dichotomy is possible in the sense of classical complexity. That is, if there are classes such that the induced subgraph isomorphism problem on is neither in nor -complete. This argument may be of independent interest, because it is applicable to various parameterized problems.
- Alon, N., Yuster, R., Zwick, U. (1995) Color-coding. J. ACM 42: pp. 844-856 CrossRef
- Bodirski, M., Grohe, M.: Non-dichotomies in Constraint Satisfaction Complexity. In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP 2008, Track B) (to appear, 2008)
- Chen, Y., Flum, J. (2007) On parameterized path and chordless path problems. Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity. IEEE Computer Society, Los Alamitos, pp. 250-263 CrossRef
- Dalmau, V., Jonsson, P. (2004) The complexity of counting homomorphisms seen from the other side. Theor. Comput. Sci. 329: pp. 315-323 CrossRef
- Dalmau, V., Kolaitis, P.G., Vardi, M.Y.: Constraint satisfaction, bounded treewidth, and finite-variable logics. In: Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming, CP 2002, pp. 310–326 (2002)
- Downey, R.G., Fellows, M.R. (1995) Fixed-parameter tractability and completeness I: Basic results. SIAM J. Comput. 24: pp. 873-921 CrossRef
- Flum, J., Grohe, M. (2001) Fixed-parameter tractability, definability, and model-checking. SIAM J. Comput. 31: pp. 113-145 CrossRef
- Flum, J., Grohe, M. (2004) The parameterized complexity of counting problems. SIAM J. Comput. 33: pp. 892-922 CrossRef
- Flum, J., Grohe, M. (2006) Parameterized Complexity Theory. Springer, Heidelberg
- Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54(1) (2007)
- Ladner, R.E. (1975) On the structure of polynomial time reducibility. J. ACM 22: pp. 155-171 CrossRef
- McCartin, C. (2006) Parameterized counting problems. Ann. Pure Appl. Logic 138: pp. 147-182 CrossRef
- Understanding the Complexity of Induced Subgraph Isomorphisms
- Book Title
- Automata, Languages and Programming
- Book Subtitle
- 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part I
- pp 587-596
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- Lecture Notes in Computer Science
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- Springer Berlin Heidelberg
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- Springer Berlin Heidelberg
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- Editor Affiliations
- 1. School of Computer Science, Reykjavik University
- 2. Department of Computer Science, IT-Parken, University of Aarhus
- 3. Department of Computer Science, University of Liverpool
- 4. School of Computer Science, Reykjavik University
- 5. Department of Computer Science, Reykjavik University
- 6. Université de Bordeaux-1, LaBRI
- Author Affiliations
- 7. BASICS, Department of Computer Science, Shanghai Jiaotong University, 200030, Shanghai, China
- 8. Institut für Informatik, Humboldt-Universität zu Berlin, 10099, Berlin, Germany
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