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I would like to know if there exists an algorithm for finding all/minimum admissible sets in a graph that satisfy the backdoor criterion as defined by Judea Pearl in his book Causality (Please see here).

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Yes. I'm not sure about the algorithm, but it is currently implemented in the dagitty tool. If you enter your proposed DAG, it will list the adjustment sets for a given total causal effect. More details are offered on the homepage about how the algorithm works, with relevant papers.

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    $\begingroup$ Noah, dagitty does not list all admissible sets, it lists the minimal sufficient sets. $\endgroup$ – Carlos Cinelli Aug 20 '18 at 20:00
  • $\begingroup$ Actually, minimal sufficient sets works also for me. My bad, I should have made the question more specific. Just my first question here. $\endgroup$ – Mauricio Soto Alvarez Aug 22 '18 at 15:38
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Yes, there are, the most recent development I am aware of is from Correa and Bareinboim, which provides:

a complete algorithm with polynomial delay to find all sets of admissible covariates for adjustment when confounding and selection biases are simultaneously present and unbiased data is available.

It finds not only for backdoor adjustment, but also considering possible selection biases.

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