# Separators and Adjustment Sets in Causal Graphs: Complete Criteria and an Algorithmic Framework

arXiv: https://arxiv.org/abs/1803.00116 · HTML: https://mlsanity.com/paper/1803.00116/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2018.

## Built on by (most cited first, 16 of 16)

- [Data-Driven Causal Effect Estimation Based on Graphical Causal Modelling: A Survey](https://mlsanity.com/paper/2208.09590/) (2022)
- [Necessary and sufficient graphical conditions for optimal adjustment sets in causal graphical models with hidden variables](https://mlsanity.com/paper/2102.10324/) (2021)
- [Polynomial-Time Algorithms for Counting and Sampling Markov Equivalent DAGs with Applications](https://mlsanity.com/paper/2205.02654/) (2022)
- [Combinatorial Pure Exploration of Causal Bandits](https://mlsanity.com/paper/2206.07883/) (2022)
- [Finding and Listing Front-door Adjustment Sets](https://mlsanity.com/paper/2210.05816/) (2022)
- [Invariant Ancestry Search](https://mlsanity.com/paper/2202.00913/) (2022)
- [Linear-Time Algorithms for Front-Door Adjustment in Causal Graphs](https://mlsanity.com/paper/2211.16468/) (2022)
- [Identifying Causal Effects Under Functional Dependencies](https://mlsanity.com/paper/2403.04919/) (2024)
- [A note on efficient minimum cost adjustment sets in causal graphical models](https://mlsanity.com/paper/2201.02037/) (2022)
- [Automating the Selection of Proxy Variables of Unmeasured Confounders](https://mlsanity.com/paper/2405.16130/) (2024)
- [Classifying Causal Structures: Ascertaining when Classical Correlations are Constrained by Inequalities](https://mlsanity.com/paper/2308.02380/) (2023)
- [Disentangled Latent Representation Learning for Tackling the Confounding M-Bias Problem in Causal Inference](https://mlsanity.com/paper/2312.05404/) (2023)
- [Leaning Time-Varying Instruments for Identifying Causal Effects in Time-Series Data](https://mlsanity.com/paper/2411.17774/) (2024)
- [Linear-Time Primitives for Algorithm Development in Graphical Causal Inference](https://mlsanity.com/paper/2506.15758/) (2025)
- [Estimate Collapsibility of Causal Effects in Completed Partial DAGs via Strong d-Convex Hulls](https://mlsanity.com/paper/2606.08941/) (2026)
- [Knowledge-Informed Local Causal Discovery of Optimal Adjustment Sets](https://mlsanity.com/paper/2607.04447/) (2026)

## Abstract (arXiv)

Principled reasoning about the identifiability of causal effects from non-experimental data is an important application of graphical causal models. This paper focuses on effects that are identifiable by covariate adjustment, a commonly used estimation approach. We present an algorithmic framework for efficiently testing, constructing, and enumerating m-separators in ancestral graphs (AGs), a class of graphical causal models that can represent uncertainty about the presence of latent confounders. Furthermore, we prove a reduction from causal effect identification by covariate adjustment to m-separation in a subgraph for directed acyclic graphs (DAGs) and maximal ancestral graphs (MAGs). Jointly, these results yield constructive criteria that characterize all adjustment sets as well as all minimal and minimum adjustment sets for identification of a desired causal effect with multivariate exposures and outcomes in the presence of latent confounding. Our results extend several existing solutions for special cases of these problems. Our efficient algorithms allowed us to empirically quantify the identifiability gap between covariate adjustment and the do-calculus in random DAGs and MAGs, covering a wide range of scenarios. Implementations of our algorithms are provided in the R package dagitty.