  Learning causal relationships between variables is a fundamental task in
causal inference and directed acyclic graphs (DAGs) are a popular choice to
represent the causal relationships. As one can recover a causal graph only up
to its Markov equivalence class from observations, interventions are often used
for the recovery task. Interventions are costly in general and it is important
to design algorithms that minimize the number of interventions performed. In
this work, we study the problem of identifying the smallest set of
interventions required to learn the causal relationships between a subset of
edges (target edges). Under the assumptions of faithfulness, causal
sufficiency, and ideal interventions, we study this problem in two settings:
when the underlying ground truth causal graph is known (subset verification)
and when it is unknown (subset search). For the subset verification problem, we
provide an efficient algorithm to compute a minimum sized interventional set;
we further extend these results to bounded size non-atomic interventions and
node-dependent interventional costs. For the subset search problem, in the
worst case, we show that no algorithm (even with adaptivity or randomization)
can achieve an approximation ratio that is asymptotically better than the
vertex cover of the target edges when compared with the subset verification
number. This result is surprising as there exists a logarithmic approximation
algorithm for the search problem when we wish to recover the whole causal
graph. To obtain our results, we prove several interesting structural
properties of interventional causal graphs that we believe have applications
beyond the subset verification/search problems studied here.
