The Complexity of Minimizing FBDDs

Free Binary Decision Diagrams (FBDDs) are a data structure for the representation and manipulation of Boolean functions. Efficient algorithms for most of the important operations are known if only FBDDs respecting a fixed graph ordering are considered. However, the size of such an FBDD may strongly depend on the chosen graph ordering and efficient algorithms for computing good or optimal graph orderings are not known. In this paper it is shown that the existence of polynomial time approximation schemes for optimizing graph orderings or for minimizing FBDDs implies NP = ZPP or NP = P, respectively, and so such algorithms are quite unlikely to exist.