# Verification and search algorithms for causal DAGs

We study two problems related to recovering causal graphs from interventional data: (i) verification, where the task is to check if a purported causal graph is correct, and (ii) search, where the task is to recover the correct causal graph. For both, we wish to minimize the number of interventions performed. For the first problem, we give a characterization of a minimal sized set of atomic interventions that is necessary and sufficient to check the correctness of a claimed causal graph. Our characterization uses the notion of covered edges, which enables us to obtain simple proofs and also easily reason about earlier results. We also generalize our results to the settings of bounded size interventions and node-dependent interventional costs. For all the above settings, we provide the first known provable algorithms for efficiently computing (near)-optimal verifying sets on general graphs. For the second problem, we give a simple adaptive algorithm based on graph separators that produces an atomic intervention set which fully orients any essential graph while using ๐ช(log n) times the optimal number of interventions needed to verify (verifying size) the underlying DAG on n vertices. This approximation is tight as any search algorithm on an essential line graph has worst case approximation ratio of ฮฉ(log n) with respect to the verifying size. With bounded size interventions, each of size โค k, our algorithm gives an ๐ช(log n ยทloglog k) factor approximation. Our result is the first known algorithm that gives a non-trivial approximation guarantee to the verifying size on general unweighted graphs and with bounded size interventions.

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