Skip to main content
SpringerLink
Log in

Canonizing Hypergraphs under Abelian Group Action

  • Conference paper
Computing and Combinatorics (COCOON 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6842))

Included in the following conference series:

Abstract

We study the problem of canonizing hypergraphs under Abelian group action and show tight complexity bounds. Our approach is algebraic. We transform the problem of graph canonization to the problem of canonizing associated algebraic structures for which we develop a parallel algorithm. Specifically, we show that the problem of computing canonical labelings for hypergraphs of color class size 2 is computable in FL  ⊕ L. For general hypergraphs, under Abelian permutation group action, for the canonization problem we show an upper bound of randomized FL GapL (which is contained in randomized NC 2). This is a nearly tight characterization since the problem is hard for the complexity class FL GapL. The problem is also in deterministic NC 3.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allender, E.: Arithmetic Circuits and Counting Complexity Classes. In: Krajicek, J., di Matematica, Q. (eds.) Complexity of Computations and Proofs. Seconda Universita di Napoli, vol. 13, pp. 33–72 (2004)

    Google Scholar 

  2. Allender, E., Ogihara, M.: Relationships among PL, #L and the determinant. RAIRO - Theoretical Informatics and Applications 30(1), 1–21 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Àlvarez, C., Jenner, B.: A Very Hard log-Space Counting Class. Theoretical Computer Science 107(1), 3–30 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arvind, V., Das, B., Mukhopadhyay, P.: On isomorphism and canonization of tournaments and hypertournaments. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 449–459. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  5. Arvind, V., Köbler, J.: Hypergraph isomorphism testing for bounded color classes. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 384–395. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  6. Arvind, V., Vijayaraghavan, T.C.: The complexity of solving linear equations over a finite ring. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 472–484. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  7. Babai, L., Luks, E.: Canonical labeling of graphs. In: Proc. 15th ACM Symposium on Theory of Computing, pp. 171–183 (1983)

    Google Scholar 

  8. Buntrock, G., Damm, C., Hertrampf, U., Meinel, C.: Structure and importance of logspace-MOD classes. Mathematical Systems Theory 25, 223–237 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Buss, S.: Alogtime algorithms for tree isomorphism, comparison, and canonization. In: Gottlob, G., Leitsch, A., Mundici, D. (eds.) KGC 1997. LNCS, vol. 1289, pp. 18–33. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  10. Furst, M., Hopcroft, J., Luks, E.: Polynomial time algorithms for permutation groups. In: Proc. 21st IEEE Symposium on the Foundations of Computer Science, pp. 36–41. IEEE Computer Society Press, Los Alamitos (1980)

    Google Scholar 

  11. Jenner, B., Köbler, J., McKenzie, P., Torán, J.: Completeness results for graph isomorphism. Journal of Computer and System Sciences 66, 549–566 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lindell, S.: A logspace algorithm for tree canonization. In: Proc. 24th ACM Symposium on Theory of Computing, pp. 400–404. ACM Press, New York (1992)

    Google Scholar 

  13. Luks, E.M.: Permutation groups and polynomial time computations. In: Finkelstein, L., Kantor, W.M. (eds.) Groups and Computation. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 11, pp. 139–175. American Mathematical Society, Providence (1993)

    Chapter  Google Scholar 

  14. Luks, E.M., McKenzie, P.: Parallel algorithms for solvable permutation groups. Journal of Computer and System Sciences 37(1), 39–62 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  15. McKenzie, P., Cook, S.A.: The parallel complexity of abelian permutation group problems. SIAM Journal on Computing 16(3), 880–909 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Miller, G.L., Reif, J.H.: Parallel tree contraction. Part 2: Further applications. SIAM Journal on Computing 20(6), 1128–1147 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Reingold, O.: Undirected st-connectivity in log-space. In: Proc. 37th ACM Symposium on Theory of Computing, pp. 376–385. ACM Press, New York (2005)

    Google Scholar 

  18. Torán, J.: On the hardness of graph isomorphism. SIAM Journal on Computing 33(5), 1093–1108 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wielandt, H.: Permutation Groups. Academic Press, New York (1964)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Institute of Mathematical Sciences, Chennai, 600 113, India

    V. Arvind

  2. Institut für Informatik, Humboldt Universität zu Berlin, Germany

    Johannes Köbler

Authors
  1. V. Arvind
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Johannes Köbler
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Texas-Pan American, 78539, Edinburg, TX, USA

    Bin Fu

  2. Department of Computer Science, University of Texas at Dallas, 75080, Richardson, TX, USA

    Ding-Zhu Du

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Arvind, V., Köbler, J. (2011). Canonizing Hypergraphs under Abelian Group Action. In: Fu, B., Du, DZ. (eds) Computing and Combinatorics. COCOON 2011. Lecture Notes in Computer Science, vol 6842. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22685-4_39

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22685-4_39

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22684-7

  • Online ISBN: 978-3-642-22685-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation