  Subgroup-discovery methods allow users to obtain simple descriptions of
interesting regions in a dataset. Using constraints in subgroup discovery can
enhance interpretability even further. In this article, we focus on two types
of constraints: First, we limit the number of features used in subgroup
descriptions, making the latter sparse. Second, we propose the novel
optimization problem of finding alternative subgroup descriptions, which cover
a similar set of data objects as a given subgroup but use different features.
We describe how to integrate both constraint types into heuristic
subgroup-discovery methods. Further, we propose a novel Satisfiability Modulo
Theories (SMT) formulation of subgroup discovery as a white-box optimization
problem, which allows solver-based search for subgroups and is open to a
variety of constraint types. Additionally, we prove that both constraint types
lead to an NP-hard optimization problem. Finally, we employ 27
binary-classification datasets to compare heuristic and solver-based search for
unconstrained and constrained subgroup discovery. We observe that heuristic
search methods often yield high-quality subgroups within a short runtime, also
in scenarios with constraints.
