Martin Grohe
Skip Abstract Section
Abstract
We give a logical characterization of the polynomial-time properties of graphs with excluded minors: For every class C of graphs such that some graph H is not a minor of any graph in C, a property P of graphs in C is decidable in polynomial time if and only if it is definable in fixed-point logic with counting. Furthermore, we prove that for every class C of graphs with excluded minors there is a k such that a simple combinatorial algorithm, namely "the k-dimensional Weisfeiler--Lehman algorithm," decides isomorphism of graphs in C in polynomial time.
- Cai, J., Fürer, M., Immerman, N. An optimal lower bound on the number of variables for graph identification. Combinatorica, 12 (1992), 389--410.Google Scholar
Cross Ref
- Chandra, A., Harel, D. Structure and complexity of relational queries. J Comput. Syst. Sci., 25 (1982), 99--128.Google ScholarCross Ref
- Fagin, R. Generalized first-order spectra and polynomial-time recognizable sets. In Complexity of Computation, SIAM--AMS Proceedings, vol. 7, R. M. Karp, ed. 1974, 43--73.Google Scholar
- Filotti, I. S., Mayer, J. N. A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus. In Proceedings of the 12th ACM Symposium on Theory of Computing, 1980, 236--243. Google ScholarDigital Library
- Grohe, M. Fixed-point logics on planar graphs. In Proceedings of the 13th IEEE Symposium on Logic in Computer Science, 1998, 6--15. Google ScholarDigital Library
- Grohe, M., Mariño, J. Definability and descriptive complexity on databases of bounded tree-width. In Proceedings of the 7th International Conference on Database Theory, Lecture Notes in Computer Science, vol. 1540. C. Beeri and P. Buneman, eds. Springer-Verlag, Jerusalem, Israel, 1999, 70--82. Google ScholarDigital Library
- Gurevich, Y. Logic and the challenge of computer science. In Current Trends in Theoretical Computer Science. E. Börger, ed. Computer Science Press, 1988, 1--57.Google Scholar
- Hella, L., Kolaitis, P., Luosto, K. Almost everywhere equivalence of logics in finite model theory. Bull. Symbol. Logic, 2 (1996), 422--443.Google ScholarCross Ref
- Hopcroft, J., Wong, J. Linear time algorithm for isomorphism of planar graphs. In Proceedings of the 6th ACM Symposium on Theory of Computing, 1974, 172--184. Google ScholarDigital Library
- Immerman, N. Relational queries computable in polynomial time (extended abstract). In Proceedings of the 14th ACM Symposium on Theory of Computing, 1982, 147--152. Google ScholarDigital Library
- Immerman, N. Expressibility as a complexity measure: Results and directions. In Proceedings of the 2nd IEEE Symposium on Structure in Complexity Theory, 1987, 194--202.Google Scholar
- Immerman, N., Lander, E. Describing graphs: A first-order approach to graph canonization. In Complexity Theory Retrospective. A. Selman, ed. Springer-Verlag, Heidelberg, 1990, 59--81.Google Scholar
- Laubner, B. Capturing polynomial time on interval graphs. In Proceedings of the 25th IEEE Symposium on Logic in Computer Science, 2010, 199--208. Google ScholarDigital Library
- Luks, E. Isomorphism of graphs of bounded valance can be tested in polynomial time. J. Comput. Syst. Sci., 25 (1982), 42--65.Google ScholarCross Ref
- Matijasevič, J. Enumerable sets are diophantic. Sov. Math. Dokl., 11 (1970), 345--357.Google Scholar
- Miller, G. L. Isomorphism testing for graphs of bounded genus. In Proceedings of the 12th ACM Symposium on Theory of Computing, 1980, 225--235. Google ScholarDigital Library
- Otto, M. Bounded Variable Logics and Counting---A Study in Finite Models, Lecture Notes in Logic, vol. 9. Springer-Verlag, Berlin, 1997.Google ScholarCross Ref
- Ponomarenko, I. N. The isomorphism problem for classes of graphs that are invariant with respect to contraction. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 174 (Teor. Slozhn. Vychisl. 3):147--177, 182, 1988 (in Russian).Google Scholar
- Robertson, N., Seymour, P. Graph minors I--XXIII. Appearing in J. Combin. Theory Ser. B since 1982.Google Scholar
- Robertson, N., Seymour, P. Graph minors XVI. Excluding a non-planar graph. J. Combin. Theory Ser. B, 77 (1999), 1--27. Google ScholarDigital Library
- Robertson, N., Seymour, P. Graph minors XX. Wagner's conjecture. J. Combin. Theory Ser. B, 92 (2004), 325--357. Google ScholarDigital Library
- Vardi, M. The complexity of relational query languages. In Proceedings of the 14th ACM Symposium on Theory of Computing, 1982, 137--146. Google ScholarDigital Library
Index Terms
- From polynomial time queries to graph structure theory
Recommendations
From polynomial time queries to graph structure theory
ICDT '10: Proceedings of the 13th International Conference on Database TheoryIn a fundamental article on query languages for relational databases, Chandra and Harel [2] asked in 1982 whether there is a language that expresses precisely those queries which can be answered in polynomial time. Gurevich [10] later rephrased the ...
Comments