Bounds on BDD-Based Bucket Elimination
This provides theoretical insights into the limitations of BDD-based satisfiability solving methods, which is incremental work for the automated reasoning community.
The paper analyzes BDD-based bucket elimination for satisfiability testing, proving it can efficiently solve pigeonhole principle formulas with specific variable order combinations, but showing this efficiency breaks down with restricted formulas or different implementation approaches.
We study BDD-based bucket elimination, an approach to satisfiability testing using variable elimination which has seen several practical implementations in the past. We prove that it allows solving the standard pigeonhole principle formulas efficiently, when allowing different orders for variable elimination and BDD-representations, a variant of bucket elimination that was recently introduced. Furthermore, we show that this upper bound is somewhat brittle as for formulas which we get from the pigeonhole principle by restriction, i.e., fixing some of the variables, the same approach with the same variable orders has exponential runtime. We also show that the more common implementation of bucket elimination using the same order for variable elimination and the BDDs has exponential runtime for the pigeonhole principle when using either of the two orders from our upper bound, which suggests that the combination of both is the key to efficiency in the setting.