Abstract
We introduce the concept of a class of graphs, or more generally, relational structures, being locally tree-decomposable. There are numerous examples of locally tree-decomposable classes, among them the class of planar graphs and all classes of bounded valence or of bounded tree-width. We also consider a slightly more general concept of a class of structures having bounded local tree-width.We show that for each property ϕ of structures that is definable in first-order logic and for each locally tree-decomposable class C of structures, there is a linear time algorithm deciding whether a given structure A ∈ C has property ϕ. For classes C of bounded local tree-width, we show that for every k ≥ 1 there is an algorithm solving the same problem in time O(n1+(1/k)) (where n is the cardinality of the input structure).
- AHO, A., HOPCROFT, J., AND ULLMAN, J. 1974. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Mass.]] Google ScholarDigital Library
- AHO, A., AND ULLMAN, J. 1979. The universality of data retrieval languages. In Proceedings of the 6th Annual ACM Symposium on Principles of Programming Languages. ACM, New York, pp. 110-120.]] Google ScholarDigital Library
- ALBER, J., FERNAU, H., AND NIEDERMEIER, R. 2001. Parameterized complexity: Exponential speed-up for planar graph problems. In Proceedings of the 28th International Colloquium on Automata; Languages and Programming (ICALP 2001). F. Orejas, P. Spirakis, and J. van Leeuwen, Eds. Lecture Notes in Computer Science, vol. 2076. Springer-Verlag, New York, pp. 261-272.]] Google ScholarDigital Library
- AWERBUCH, B., BERGER, B., COWEN, L., AND PELEG, D. 1993. Near-linear cost sequential and distributed onstructions of sparse neighborhood covers. In Proceedings of the 34th Annual IEEE Sym-posium on Foundations of Computer Science. IEEE Computer Science Press, Los Alamitos, Calif., pp. 638-647.]]Google Scholar
- AWERBUCH, B., AND PELEG, D. 1990. Sparse partitions. In Proceedings of the 31st Annual IEEE Symposium on Foundations of Computer Science. IEEE Computer Science Press, Los Alamitos, Calif., pp. 503-513.]]Google ScholarDigital Library
- BAKER, B. 1994. Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153-180.]] Google ScholarDigital Library
- BODLAENDER, H. 1996. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305-1317.]] Google ScholarDigital Library
- BODLAENDER, H. 1997. Treewidth: Algorithmic techniques and results. In Proceedings of the 22nd International Symposium on Mathematical Foundations of Computer Science; MFCS'97; I. Privara and P. Ruzicka, Eds. Lecture Notes in Computer Science, vol. 1295. Springer-Verlag, New York, pp. 29-36.]] Google ScholarDigital Library
- COURCELLE, B. 1990. Graph rewriting: An algebraic and logic approach. In Handbook of Theoretical Computer Science, J. van Leeuwen, Ed. Vol. 2. Elsevier Science Publishers, Amsterdam, The Netherlands, pp. 194-242.]] Google ScholarDigital Library
- COURCELLE, B., ENGELFRIET, J., AND ROZENBERG, G. 1993. Handle-rewriting hypergraph grammars. J. Comput. Syst. Sci. 46, 218-270.]] Google ScholarDigital Library
- COURCELLE, B., MAKOWSKY, J., AND ROTICS, U. 2000. Linear time solvable optimization problems on graphs of bounded clique width. Theory Comput. Syst. 33; 2, 125-150.]]Google Scholar
- DIESTEL, R. 2000. Graph Theory, Second ed. Springer-Verlag, New York.]]Google Scholar
- DOWNEY, R., AND FELLOWS, M. 1999. Parameterized Complexity. Springer-Verlag, New York.]]Google Scholar
- DOWNEY, R., FELLOWS, M., AND TAYLOR, U. 1996. The parameterized complexity of relational database queries and an improved characterization of W{1}. In Combinatorics; Complexity; and Logic- Proceedings of DMTCS '96. D. S. Bridges, C. Calude, P. Gibbons, S. P. Reeves, and I. H. Witten, Eds. Springer-Verlag, New York, pp. 194-213.]]Google Scholar
- EPPSTEIN, D. 1999. Subgraph isomorphism in planar graphs and related problems. J. Graph Algor. Appli. 3, 1-27.]]Google ScholarCross Ref
- EPPSTEIN, D. 2000. Diameter and treewidth in minor-closed graph families. Algorithmica 27, 275-291.]]Google ScholarCross Ref
- ERDOS, P. 1959. Graph theory and probability. Canadi. J. Math. 11, 34-38.]]Google Scholar
- FLUM, J., FRICK, M., AND GROHE, M. 2001. Query evaluation via tree-decompositions. Currently available at http://www.dcs.ed.ac.uk/grohe. A preliminary version of the paper appeared. In Proceedings of the 8th International Conference on Database Theory, LNCS 1973, Springer-Verlag.]] Google ScholarDigital Library
- FLUM, J., AND GROHE, M. 2001. Fixed-parameter tractability, definability, and model checking. SIAM J. Comput. 31; 1, 113-145.]] Google ScholarDigital Library
- GAIFMAN, H. 1982. On local and non-local properties. In Proceedings of the Herbrand Symposium; Logic Colloquium '81. North-Holland, Amsterdam, The Netherlands.]]Google ScholarCross Ref
- GAREY, M., JOHNSON, D., AND STOCKMEYER, L. 1976. Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1, 237-267.]]Google ScholarCross Ref
- GROHE, M. 2001. Local tree-width, excluded minors, and approximation algorithms. Combinatorica. To appear.]] Google ScholarDigital Library
- GROHE, M., AND WOHRLE, S. 2001. An existential locality theorem. In Proceeding of the Computer Science Logic; 15th International Workshop CSL'01. Lecture Notes in Computer Science, vol. 2142. Springer-Verlag, New York, pp. 99-114.]] Google ScholarDigital Library
- IMMERMAN, N. 1982. Upper and lower bounds for first-order expressibility. J. Comput. Syst. Sci. 25, 76-98.]]Google ScholarCross Ref
- LIFSCHES, S., AND SHELAH, S. 2001. Distorted sums of models. Unpublished manuscript.]]Google Scholar
- PAPADIMITRIOU, C., AND YANNAKAKIS, M. 1997. On the complexity of database queries. In Proceedings of the 17th ACM Symposium on Principles of Database Systems. ACM, New York, pp. 12-19.]] Google ScholarDigital Library
- PELEG, D. 1993. Distance-dependent distributed directories. Info. Computa. 103, 270-298.]] Google ScholarDigital Library
- ROBERTSON, N., AND SEYMOUR, P. 1984. Graph minors III. Planar tree-width. J. Combinat. Theory; Ser. B 36, 49-64.]]Google Scholar
- ROBERTSON, N., AND SEYMOUR, P. 1986. Graph minors V. Excluding a planar graph. J. Combinat. Theory; Ser. B 41, 92-114.]] Google ScholarDigital Library
- ROBERTSON, N., AND SEYMOUR, P. 1995. Graph minors XIII. The disjoint paths problem. J. Combinat. Theory; Ser. B 63, 65-110.]] Google ScholarDigital Library
- SEESE, D. 1996. Linear time computable problems and first-order descriptions. Math. Structu. Comput. Sci. 6, 505-526.]]Google ScholarCross Ref
- THOMASSEN, C. 1995. Embeddings and minors. In Handbook of Combinatorics, R. Graham, M. Grotschel, and L. Lovasz, Eds. Vol. 1, Chap. 5. Elsevier, Amsterdam, The Netherlands, pp. 301-349.]] Google ScholarDigital Library
- VAN EMDE BOAS, P. 1990. Machine models and simulations. In Handbook of Theoretical Computer Science, J. van Leeuwen, Ed. Vol. 1. Elsevier Science Publishers, Amsterdam, The Netherlands, pp. 1-66.]] Google ScholarDigital Library
- VARDI, M. 1982. The complexity of relational query languages. In Proceedings of the 14th ACM Symposium on Theory of Computing. ACM, New York, pp. 137-146.]] Google ScholarDigital Library
- YANNAKAKIS, M. 1995. Perspectives on database theory. In Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, New York, pp. 224-246.]] Google ScholarDigital Library
Index Terms
- Deciding first-order properties of locally tree-decomposable structures
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