We study the problem of learning the causal relationships between a set of observed variables in the presence of latents, while minimizing the cost of interventions on the observed variables. We assume access to an undirected graph $G$ on the observed variables whose edges represent either all direct causal relationships or, less restrictively, a superset of causal relationships (identified, e.g., via conditional independence tests or a domain expert). Our goal is to recover the directions of all causal or ancestral relations in $G$, via a minimum cost set of interventions. It is known that constructing an exact minimum cost intervention set for an arbitrary graph $G$ is NP-hard. We further argue that, conditioned on the hardness of approximate graph coloring, no polynomial time algorithm can achieve an approximation factor better than $Θ(\log n)$, where $n$ is the number of observed variables in $G$. To overcome this limitation, we introduce a bi-criteria approximation goal that lets us recover the directions of all but $εn^2$ edges in $G$, for some specified error parameter $ε> 0$. Under this relaxed goal, we give polynomial time algorithms that achieve intervention cost within a small constant factor of the optimal. Our algorithms combine work on efficient intervention design and the design of low-cost separating set systems, with ideas from the literature on graph property testing.