  We consider recovering a causal graph in presence of latent variables, where
we seek to minimize the cost of interventions used in the recovery process. We
consider two intervention cost models: (1) a linear cost model where the cost
of an intervention on a subset of variables has a linear form, and (2) an
identity cost model where the cost of an intervention is the same, regardless
of what variables it is on, i.e., the goal is just to minimize the number of
interventions. Under the linear cost model, we give an algorithm to identify
the ancestral relations of the underlying causal graph, achieving within a
$2$-factor of the optimal intervention cost. This approximation factor can be
improved to $1+\epsilon$ for any $\epsilon > 0$ under some mild restrictions.
Under the identity cost model, we bound the number of interventions needed to
recover the entire causal graph, including the latent variables, using a
parameterization of the causal graph through a special type of colliders. In
particular, we introduce the notion of $p$-colliders, that are colliders
between pair of nodes arising from a specific type of conditioning in the
causal graph, and provide an upper bound on the number of interventions as a
function of the maximum number of $p$-colliders between any two nodes in the
causal graph.
