On the complexity of some ordering problems

Abstract

Two different ordering problems are investigated. Ordered binary decision diagrams (OBDDs) are a popular data structure for Boolean functions. Some applications work with a restricted variant called complete OBDDs. This model has also been investigated in complexity theory, e.g., in property testing. It is well-known that the size of an OBDD for the representation of a given function may depend significantly on the chosen variable ordering but the computation of an optimal ordering is NP-hard. Since optimal variable orderings for OBDDs are not necessarily optimal for the complete model, the complexity to find an optimal variable ordering for complete OBDDs is investigated. Here, using a new reduction idea it is shown that the problem is NP-hard. Among the many areas of applications OBDDs have been used in the design and analysis of implicit graph algorithms where the choice of a good vertex encoding is of additional importance to represent a given input graph in small size. The computational complexity of the vertex encoding problem is unknown but in the paper a first step is done to determine its complexity by showing that a restricted case is NP-hard. © 2014 Springer-Verlag Berlin Heidelberg.