Theory of Computing Systems

, Volume 59, Issue 3, pp 532–559 | Cite as

On the Minimization of (Complete) Ordered Binary Decision Diagrams

  • Beate BolligEmail author


Ordered binary decision diagrams (OBDDs) are a popular data structure for Boolean functions. Some applications work with a restricted variant called complete OBDDs which is strongly related to nonuniform deterministic finite automata. One of its complexity measures is the width which has been investigated in several areas in computer science like machine learning, property testing, and the design and analysis of implicit graph algorithms. For a given function the size and the width of a (complete) OBDD is very sensitive to the choice of the variable ordering but the computation of an optimal variable ordering for the OBDD size is known to be NP-hard. Since optimal variable orderings with respect to the OBDD size are not necessarily optimal for the complete model or the OBDD width and hardly anything about the relation between optimal variable orderings with respect to the size and the width is known, this relationship is investigated. Here, using a new reduction idea it is shown that the size minimization problem for complete OBDDs and the width minimization problem are NP-hard.


Complexity theory Layout problems Nonuniform finite automata NP-completeness Ordered binary decision diagrams 



The author would like to thank the Deutsche Forschungsgemeinschaft (DFG) for funding the project Design and Analysis of Implicit OBDD-based Graph Algorithms (BO 2755/1).

Furthermore, we would like to thank Suzana Mitkovska for the technical realization of the figures in the paper.


  1. 1.
    Bollig, B.: On the size of (generalized) OBDDs for threshold functions. Inf. Process. Lett. 109(10), 499–503 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bollig, B.: On the complexity of some ordering problems. In: Proc. of MFCS, LNCS 8635, pp. 118–129. Springer (2014)Google Scholar
  3. 3.
    Bollig, B.: On the width of ordered binary decision diagrams. In: Proc. of COCOA, LNCS 8881, pp. 444–458. Springer (2014)Google Scholar
  4. 4.
    Bollig, B., Range, N., Wegener, I.: Exact OBDD bounds for some fundamental functions. Theory Comput. Syst. 47(2), 593–609 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bollig, B., Wegener, I.: Improving the variable ordering of OBDDs is NP-complete. IEEE Trans. Comput. 45(9), 993–1002 (1996)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bollig, B., Wegener, I.: Asymptotically optimal bounds for OBDDs and the solution of some basic OBDD problems. J. Comput. Syst. Sci. 61(3), 558–579 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brody, J., Matulef, K., Wu, C.: Lower bounds for testing computability by small width OBDDs. In: TAMC, LNCS 6648, pp. 320–331. Springer (2011)Google Scholar
  8. 8.
    Bryant, R.: Graph-based algorithms for boolean function manipulation. IEEE Trans. Comput. 35(8), 677–691 (1986)CrossRefzbMATHGoogle Scholar
  9. 9.
    Díaz, J., Petit, J., Serna, M.J.: A survey of graph layout problems. ACM Comput. Surv. 34(3), 313–356 (2002)CrossRefGoogle Scholar
  10. 10.
    Diestel, R.: Graph Theory, 4th Edition, Graduate texts in mathematics, vol. 173. Springer (2012)Google Scholar
  11. 11.
    Ergün, F., Kumar, R., Rubinfeld, R.: On learning bounded-width branching programs. In: COLT, pp. 361–368 (1995)Google Scholar
  12. 12.
    Fortune, F., Hopcroft, J., Schmidt, E.: The complexity of equivalence and containment for free single variable program schemes. In: Proc. of ICALP, LNCS 62, pp. 227–240. Springer (1978)Google Scholar
  13. 13.
    Gavril, F.: Some NP-complete problems on graphs. In: 11th Conference on Information Science and Systems, pp. 91–95 (1977)Google Scholar
  14. 14.
    Goldreich, O.: On testing computability by small width OBDDs. In: APPROX-RANDOM, pp. 574–587 (2010)Google Scholar
  15. 15.
    Newman, I.: Testing membership in languages that have small width branching programs. SIAM J. Comput. 31(5), 1557–1570 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ochi, H., Ishiura, N., Yajima, S.: Breadth-first manipulation of SBDD of boolean functions for vector processing. In: DAC, pp. 413–416 (1991)Google Scholar
  17. 17.
    Ochi, H., Yasuoka, K., Yajima, S.: Breadth-first manipulation of very large binary-decision diagrams. In: ICCAD, pp. 48–55 (1993)Google Scholar
  18. 18.
    Ron, D., Tsur, G.: Testing computability by width-two OBDDs. Theor. Comput. Sci. 420, 64–79 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sawitzki, D.: The complexity of problems on implicitly represented inputs. In: Proc. of SOFSEM, LNCS 3831, pp. 471–482. Springer (2006)Google Scholar
  20. 20.
    Sieling, D.: The nonapproximability of OBDD minimization. Inf. Comput. 172(2), 103–138 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sieling, D., Wegener, I.: NC-algorithms for operations on binary decision diagrams. Parallel Process. Lett. 3, 3–12 (1993)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tani, S., Hamagushi, K., Yajima, S.: The complexity of the optimal variable ordering problems of a shared binary decision diagram. In: Proc. of ISAAC, LNCS 762, pp. 389–396. Springer (1993)Google Scholar
  23. 23.
    Wegener, I.: The size of reduced OBDDs and optimal read-once branching programs for almost all boolean functions. IEEE Trans. Comput. 43(11), 1262–1269 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wegener, I.: Branching programs and binary decision diagrams: theory and applications. SIAM (2000)Google Scholar
  25. 25.
    Woelfel, P.: Symbolic topological sorting with OBDDs. J. Discret. Algoritm. 4(1), 51–71 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.TU Dortmund, LS 2 InformatikDortmundGermany

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