Abstract:
Free Binary Decision Diagrams (FBDDs) or read-once branching programs are a data structure for Boolean functions. They can efficiently be manipulated if only FBDDs respecting a fixed graph ordering are considered. However, the size of such FBDDs may strongly depend on the chosen graph ordering. In this paper it is shown that the existence of polynomial time approximation schemes for optimizing graph orderings or for minimizing FBDDs implies NP=P, and so such algorithms are quite unlikely to exist. The same holds for the related problem of computing minimal size FBDDs that are consistent with a given set of examples. The latter result implies that size bounded FBDDs are not PAC-learnable unless NP=RP.

Abstract:
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.