Finding Lean Induced Cycles in Binary Hypercubes

  • Yury Chebiryak
  • Thomas Wahl
  • Daniel Kroening
  • Leopold Haller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5584)


Induced (chord-free) cycles in binary hypercubes have many applications in computer science. The state of the art for computing such cycles relies on genetic algorithms, which are, however, unable to perform a complete search. In this paper, we propose an approach to finding a special class of induced cycles we call lean, based on an efficient propositional SAT encoding. Lean induced cycles dominate a minimum number of hypercube nodes. Such cycles have been identified in Systems Biology as candidates for stable trajectories of gene regulatory networks. The encoding enabled us to compute lean induced cycles for hypercubes up to dimension 7. We also classify the induced cycles by the number of nodes they fail to dominate, using a custom-built All-SAT solver. We demonstrate how clause filtering can reduce the number of blocking clauses by two orders of magnitude.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdulla, P.A., Iyer, S.P., Nylén, A.: SAT-solving the coverability problem for Petri nets. Formal Methods in System Design 24(1), 25–43 (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chebiryak, Y., Kroening, D.: An efficient SAT encoding of circuit codes. In: Procs. IEEE International Symposium on Information Theory and its Applications, Auckland, New Zealand, December 2008, pp. 1235–1238 (2008)Google Scholar
  3. 3.
    Chebiryak, Y., Kroening, D.: Towards a classification of Hamiltonian cycles in the 6-cube. Journal on Satisfiability, Boolean Modeling and Computation (JSAT) 4, 57–74 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    de Jong, H., Page, M.: Search for steady states of piecewise-linear differential equation models of genetic regulatory networks. IEEE/ACM Trans. Comput. Biology Bioinform. 5(2), 208–222 (2008)CrossRefGoogle Scholar
  5. 5.
    Diaz-Gomez, P.A., Hougen, D.F.: Genetic algorithms for hunting snakes in hypercubes: Fitness function analysis and open questions. In: SNPD-SAWN 2006: Proceedings of the Seventh ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing, Washington, DC, USA, pp. 389–394. IEEE Computer Society, Los Alamitos (2006)Google Scholar
  6. 6.
    Dransfield, M.R., Marek, V.W., Truszczynski, M.: Satisfiability and computing van der Waerden numbers. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 1–13. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Edwards, R.: Symbolic dynamics and computation in model gene networks. Chaos 11(1), 160–169 (2001)CrossRefGoogle Scholar
  8. 8.
    Eén, N., Biere, A.: Effective preprocessing in SAT through variable and clause elimination. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 61–75. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Glass, L.: Combinatorial aspects of dynamics in biological systems. In: Landman, U. (ed.) Statistical mechanics and statistical methods in theory and applications, pp. 585–611. Plenum Press (1977)Google Scholar
  11. 11.
    Knuth, D.E.: The Art of Computer Programming. fascicle 2: Generating All Tuples and Permutations, vol. 4. Addison-Wesley Professional, Reading (2005)zbMATHGoogle Scholar
  12. 12.
    Kouril, M., Franco, J.V.: Resolution tunnels for improved SAT solver performance. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 143–157. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Liu, X., Schrack, G.F.: A heuristic approach for constructing symmetric Gray codes. Appl. Math. Comput. 155(1), 55–63 (2004)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Livingston, M., Stout, Q.: Perfect dominating sets. Congressus Numerantium 79, 187–203 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Na’aman Kam, D., Kugler, H., Rami Marelly, A., Hubbard, J., Stern, M.: Formal modelling of C. elegans development. A scenario-based approach. Modelling in Molecular Biology, 151–174 (2004)Google Scholar
  16. 16.
    Pipatsrisawat, K., Darwiche, A.: A lightweight component caching scheme for satisfiability solvers. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 294–299. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Savage, C.: A survey of combinatorial Gray codes. SIAM Review 39(4), 605–629 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schewe, L.: Generation of oriented matroids using satisfiability solvers. In: Iglesias, A., Takayama, N. (eds.) ICMS 2006. LNCS, vol. 4151, pp. 216–218. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Singleton, R.C.: Generalized snake-in-the-box codes. IEEE Transactions on Electronic Computers EC-15(4), 596–602 (1966)CrossRefGoogle Scholar
  20. 20.
    Zinovik, I., Chebiryak, Y., Kroening, D.: Cyclic attractors in Glass models for gene regulatory networks. IEEE Trans. Inf. Theory: Special Issue on Molecular Biology and Neuroscience (December 2009) (accepted)Google Scholar
  21. 21.
    Zinovik, I., Kroening, D., Chebiryak, Y.: An algebraic algorithm for the identification of Glass networks with periodic orbits along cyclic attractors. In: Anai, H., Horimoto, K., Kutsia, T. (eds.) Ab 2007. LNCS, vol. 4545, pp. 140–154. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  22. 22.
    Zinovik, I., Kroening, D., Chebiryak, Y.: Computing binary combinatorial Gray codes via exhaustive search with SAT-solvers. IEEE Transactions on Information Theory 54(4), 1819–1823 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Yury Chebiryak
    • 1
  • Thomas Wahl
    • 1
    • 2
  • Daniel Kroening
    • 1
    • 2
  • Leopold Haller
    • 2
  1. 1.Computer Systems InstituteETH ZurichSwitzerland
  2. 2.Computing LaboratoryOxford UniversityUnited Kingdom

Personalised recommendations