Minimum intervention cover of a causal graph

Eliciting causal effects from interventions and observations is one of the central concerns of science, and increasingly, artificial intelligence. We provide an algorithm that, given a causal graph G, determines MIC(G), a minimum intervention cover of G, i.e., a minimum set of interventions that suffices for identifying every causal effect that is identifiable in a causal model characterized by G. We establish the completeness of do-calculus for computing MIC(G). MIC(G) effectively offers an efficient compilation of all of the information obtainable from all possible interventions in a causal model characterized by G. Minimum intervention cover finds applications in a variety of contexts including counterfactual inference, and generalizing causal effects across experimental settings. We analyze the computational complexity of minimum intervention cover and identify some special cases of practical interest in which MIC(G) can be computed in time that is polynomial in the size of G.