Counting Problems and the Inclusion-Exclusion Principle

In this paper, we present improved techniques for computing and getting bounds on the cardinality of a union of sets using the inclusion-exclusion principle and Bonferroni inequalities. We organize the terms involved in the inclusionexclusion sum as a tree, showing that a set inclusion between a parent and its children yields a cancellation, where we may prune an entire subtree. Next, we provide a straightforward extension to the standard Bonferroni inequalities where we obtain upper and lower bounds by pruning the tree of terms at arbitrary odd and even depths, respectively. We conclude by showing how our work can be applied to the problem of counting the number of solutions to a given propositional SAT formula.