Front-door Reducibility: Reducing ADMGs to the Standard Front-door Setting via a Graphical Criterion

Front-door adjustment gives a simple closed-form identification formula under the classical front-door criterion, but its applicability is often viewed as narrow. By contrast, the general ID algorithm can identify many more causal effects in arbitrary graphs, yet typically outputs algebraically complex expressions that are hard to estimate and interpret. We show that many such graphs can in fact be reduced to a standard front-door setting via front-door reducibility (FDR), a graphical condition on acyclic directed mixed graphs that aggregates variables into super-nodes $(\boldsymbol{X}^{*},\boldsymbol{Y}^{*},\boldsymbol{M}^{*})$. We characterize the FDR criterion, prove it is equivalent (at the graph level) to the existence of an FDR adjustment, and present FDR-TID, an exact algorithm that finds an admissible FDR triple with correctness, completeness, and finite-termination guarantees. Empirical examples show that many graphs far outside the textbook front-door setting are FDR, yielding simple, estimable adjustments where general ID expressions would be cumbersome. FDR therefore complements existing identification methods by prioritizing interpretability and computational simplicity without sacrificing generality across mixed graphs.