Advertisement

Succinct Encodings of Graph Isomorphism

  • Bireswar Das
  • Patrick Scharpfenecker
  • Jacobo Torán
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8370)

Abstract

It is well known that problems encoded with circuits or formulas generally gain an exponential complexity blow-up compared to their original complexity.

We introduce a new way for encoding graph problems, based on CNF or DNF formulas. We show that contrary to the other existing succinct models, there are examples of problems whose complexity does not increase when encoded in the new form, or increases to an intermediate complexity class less powerful than the exponential blow up.

We also study the complexity of the succinct versions of the Graph Isomorphism problem. We show that all the versions are hard for PSPACE. Although the exact complexity of these problems is not known, we show that under most existing succinct models the different versions of the problem are equivalent. We also give an algorithm for the DNF encoded version of GI whose running time depends only on the size of the succinct representation.

Keywords

Complexity Succinct Graphisomorphism CNF DNF 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Babai, L., Luks, E.M.: Canonical labeling of graphs. In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, STOC 1983, pp. 171–183. ACM Press, New York (1983)CrossRefGoogle Scholar
  2. 2.
    Balcázar, J.L., Lozano, A., Torán, J.: The Complexity of Algorithmic Problems on Succinct Instances. Computer Science, Research and Applications. Springer US (1992)Google Scholar
  3. 3.
    Eiter, T., Gottlob, G., Mannila, H.: Adding disjunction to datalog (extended abstract). In: Proceedings of the Thirteenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, PODS 1994, pp. 267–278. ACM Press, New York (1994)CrossRefGoogle Scholar
  4. 4.
    Feigenbaum, J., Kannan, S., Vardi, M.Y., Viswanathan, M.: Complexity of Problems on Graphs Represented as OBDDs. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 216–226. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  5. 5.
    Fleischner, H., Mujuni, E., Paulusma, D., Szeider, S.: Covering Graphs with Few Complete Bipartite Subgraphs. In: Arvind, V., Prasad, S. (eds.) FSTTCS 2007. LNCS, vol. 4855, pp. 340–351. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Fortnow, L., Reingold, N.: PP Is Closed under Truth-Table Reductions. Information and Computation 124(1), 1–6 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Galota, M., Vollmer, H.: Functions computable in polynomial space. Information and Computation 198(1), 56–70 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Galperin, H., Wigderson, A.: Succinct representations of graphs. Information and Control 56(3), 183–198 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Jahanjou, H., Miles, E., Viola, E.: Local reductions (2013)Google Scholar
  10. 10.
    Köbler, J., Schöning, U., Torán, J.: The graph isomorphism problem: its structural complexity. Birkhauser (August 1994)Google Scholar
  11. 11.
    Papadimitriou, C.H., Yannakakis, M.: A note on succinct representations of graphs. Information and Control 71(3), 181–185 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Schöning, U., Torán, J.: The Satisfiability Problem: Algorithms and Analyses. Lehmanns Media (2013)Google Scholar
  13. 13.
    Toran, J.: On the hardness of graph isomorphism. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 180–186 (2000)Google Scholar
  14. 14.
    Torán, J.: Reductions to Graph Isomorphism. Theory of Computing Systems 47(1), 288–299 (2008)CrossRefGoogle Scholar
  15. 15.
    Veith, H.: Languages represented by Boolean formulas. Information Processing Letters 63(5), 251–256 (1997)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Veith, H.: How to encode a logical structure by an OBDD. In: Proceedings of the 13th IEEE Conference on Computational Complexity, pp. 122–131. IEEE Comput. Soc. (1998)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Bireswar Das
    • 1
  • Patrick Scharpfenecker
    • 2
  • Jacobo Torán
    • 2
  1. 1.Indian Institute of TechnologyGandhinagarIndia
  2. 2.Institute of Theoretical Computer ScienceUniversity of UlmGermany

Personalised recommendations