  The unit selection problem aims to identify objects, called units, that are
most likely to exhibit a desired mode of behavior when subjected to stimuli
(e.g., customers who are about to churn but would change their mind if
encouraged). Unit selection with counterfactual objective functions was
introduced relatively recently with existing work focusing on bounding a
specific class of objective functions, called the benefit functions, based on
observational and interventional data -- assuming a fully specified model is
not available to evaluate these functions. We complement this line of work by
proposing the first exact algorithm for finding optimal units given a broad
class of causal objective functions and a fully specified structural causal
model (SCM). We show that unit selection under this class of objective
functions is $\text{NP}^\text{PP}$-complete but is $\text{NP}$-complete when
unit variables correspond to all exogenous variables in the SCM. We also
provide treewidth-based complexity bounds on our proposed algorithm while
relating it to a well-known algorithm for Maximum a Posteriori (MAP) inference.
