On the complexity of constructing optimal ordered binary decision diagrams

Abstract

In the following we prove the widely assumed conjecture that the construction of an optimal OBDD-representation of a Boolean function f is intractable if f is given in terms of DNFs, CNFs, logical circuits, or Ω-BDDs. The problem remains NP-hard if one wants to construct an optimal OBDD-representation merely in such cases its size is bounded by a given constant. Both results remain true if the variable ordering of an optimal OBDD-representation is known in advance.

Reflecting these theoretical results and the requirements in the area of practical applications (above all in CAD of ciruits), the problem of construction of the minimal OBDD for a given variable ordering becomes of growing importance. We show that, starting with a free BDD (FBDD) P of a Boolean function f on X, for each ordering π of X, the minimal π OBDD- representation Q of f can be constructed output-efficiently, i.e. in time that is polynomial in the sizes of input and output. This result can be considerably improved in the case of starting with an OBDD instead of an FBDD. The time requirement is then smaller than the product of input length and the square of output length (worst case), and almost the product of input and output length in average.