In several different applications, including data transformation and entity
resolution, rules are used to capture aspects of knowledge about the
application at hand. Often, a large set of such rules is generated
automatically or semi-automatically, and the challenge is to refine the
encapsulated knowledge by selecting a subset of rules based on the expected
operational behavior of the rules on available data. In this paper, we carry
out a systematic complexity-theoretic investigation of the following rule
selection problem: given a set of rules specified by Horn formulas, and a pair
of an input database and an output database, find a subset of the rules that
minimizes the total error, that is, the number of false positive and false
negative errors arising from the selected rules. We first establish
computational hardness results for the decision problems underlying this
minimization problem, as well as upper and lower bounds for its
approximability. We then investigate a bi-objective optimization version of the
rule selection problem in which both the total error and the size of the
selected rules are taken into account. We show that testing for membership in
the Pareto front of this bi-objective optimization problem is DP-complete.
Finally, we show that a similar DP-completeness result holds for a bi-level
optimization version of the rule selection problem, where one minimizes first
the total error and then the size
