Linear (causal) relationships between random variables can be conveniently encoded using a mixed graph (a graph with both directed and bidirected edges) where a directed edge implies a direct linear effect and a bidirected edge captures the existence of unobserved confounding. Even when there is a known a mixed graph that accurately reflects the data generating mechanism, that is, all causal relationships are known and linear, confounding can make it impossible to infer parameters of interest. More concretely, many mixed graphs have (generically) unidentifiable parameters. Worse yet, while some sufficient and necessary conditions exist, there is still no general criterion that certifies when such inference is, or is not, possible for a given parameter in a mixed graph. In this talk, we introduce a collection of combinatorial conditions on mixed graphs that extend all currently known sufficient conditions for identifiability. In particular, we will show how solving a collection of max-flow problems can produce invertible linear equations whose coefficients are subdeterminants of the population covariance matrix. These conditions can be thought of as "determinantal" generalizations of instrumental variables. While mixed graphs allow us to model fine-grained dependence structure, the coarser problem of testing whether two random vectors are independent is also interesting. Addressing this problem, we will briefly discuss the computation and asymptotic properties of consistent, nonparametric measures of dependence related to Kendall's tau.