Single Family Algebra Operation on ZDDs Leads To Exponential Blow-Up

Zero-suppressed binary decision diagram (ZDD) is a data structure to represent a family of (sub)sets compactly, and it can be used as a succinct index for a family of sets. To build ZDD representing a desired family of sets, there are many transformation operations that take ZDDs as inputs and output ZDD representing the resultant family after performing operations such as set union and intersection. However, except for some basic operations, the worst-time complexity of taking such transformation on ZDDs has not been extensively studied, and some contradictory statements about it have arisen in the literature. In this paper, we show that many transformation operations on ZDDs cannot be performed in worst-case polynomial time with respect to the size of input ZDDs. This refutes some of the folklore circulated in past literature and resolves an open problem raised by Knuth. Our results are stronger in that such blow-up of computational time occurs even when the ordering, which has a significant impact on the efficiency of treating ZDDs, is reasonable.