# On the Minimization of (Complete) Ordered Binary Decision Diagrams

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## Abstract

Ordered binary decision diagrams (OBDDs) are a popular data structure for Boolean functions. Some applications work with a restricted variant called complete OBDDs which is strongly related to nonuniform deterministic finite automata. One of its complexity measures is the width which has been investigated in several areas in computer science like machine learning, property testing, and the design and analysis of implicit graph algorithms. For a given function the size and the width of a (complete) OBDD is very sensitive to the choice of the variable ordering but the computation of an optimal variable ordering for the OBDD size is known to be NP-hard. Since optimal variable orderings with respect to the OBDD size are not necessarily optimal for the complete model or the OBDD width and hardly anything about the relation between optimal variable orderings with respect to the size and the width is known, this relationship is investigated. Here, using a new reduction idea it is shown that the size minimization problem for complete OBDDs and the width minimization problem are NP-hard.

## Keywords

Complexity theory Layout problems Nonuniform finite automata NP-completeness Ordered binary decision diagrams## Notes

### Acknowledgments

The author would like to thank the *Deutsche Forschungsgemeinschaft* (DFG) for funding the project *Design and Analysis of Implicit OBDD-based Graph Algorithms* (BO 2755/1).

Furthermore, we would like to thank Suzana Mitkovska for the technical realization of the figures in the paper.

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