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.

翻译：零抑制二元决策图（ZDD）是一种用于紧凑表示（子）集族的数据结构，可作为集族的简洁索引。为构建表示目标集族的ZDD，存在多种变换操作，这些操作以ZDD为输入，输出执行并集、交集等集合运算后所得结果族的ZDD。然而，除基本运算外，ZDD上此类变换的最坏时间复杂度尚未得到充分研究，文献中甚至出现矛盾结论。本文证明，ZDD上的许多变换操作无法在输入ZDD规模的多项式时间内完成最坏情况计算。这一结果反驳了过往文献中流传的部分非正式结论，并解决了Knuth提出的开放问题。我们的结论更具普适性：即使在对ZDD处理效率影响重大的变量序合理的情况下，仍会发生计算时间的爆炸性增长。