SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In this work, we introduced Meek separators. In particular, we established the existence of a small-sized Meek separator and presented efficient algorithms to compute it. Meek separators hold great potential for designing divide-and-conquer strategies to tackle various causal discovery problems. We demonstrated this by designing efficient approximation algorithms for two important problems in targeted causal discovery: subset search and causal mean matching. Our approximation guarantees are exponentially better than the guarantees achievable by any deterministic algorithm for both problems. It would be an interesting future research endeavour to explore the application of Meek separators to address other problems in the field of causal discovery.
Limitations and Future Work. We made several standard assumptions such as causal sufficiency and we considered the noiseless setting. In future work, it would be of interest to relax some of these assumptions. Particularly, we believe that investigating the sample complexity for conducting targeted causal discovery would be an important avenue to pursue.
In addition to these broad questions, there are some specific open problems. One such problem is understanding the weighted subset search problem. Although efficient algorithms have been proposed in [CS23] to compute weighted subset verification numbers, the weighted subset search problem remains open. Exploring its approximability would be an interesting research direction. Moreover, for the causal mean matching problem, extending the matching criteria beyond the mean to encompass other higher-order moments of the distribution would be a natural and compelling future direction.
The authors were supported by the Eric and Wendy Schmidt Center at the Broad Institute, as well as NCCIH/NIH (1DP2AT012345), ONR (N00014-22-1-2116), DOE-ASCR (DE-SC0023187), the MIT-IBM Watson AI Lab, and a Simons Investigator Award. J.Z. was partially supported by an Apple AI/ML PhD Fellowship.
[AB02] Réka Albert and Albert-László Barabási. Statistical mechanics of complex networks. Reviews of modern physics, 74(1):47, 2002.
[AMP97] Steen A. Andersson, David Madigan, and Michael D. Perlman. A characterization of Markov equivalence classes for acyclic digraphs. The Annals of Statistics, 25(2):505-541, 1997.
[ASY+19] Raj Agrawal, Chandler Squires, Karren Yang, Karthikeyan Shanmugam, and Caroline Uhler. Abcd-strategy: Budgeted experimental design for targeted causal structure discovery. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 3400-3409. PMLR, 2019.
[ATS03] Constantin F Aliferis, Ioannis Tsamardinos, and Alexander Statnikov. Hiton: a novel markov blanket algorithm for optimal variable selection. In AMIA annual symposium proceedings, volume 2003, page 21. American Medical Informatics Association, 2003.
[BP93] Jean R. S. Blair and Barry W. Peyton. An introduction to chordal graphs and clique trees. In Graph theory and sparse matrix computation, pages 1-29. Springer, 1993.
[CD12] Anne BC Cherry and George Q Daley. Reprogramming cellular identity for regenerative medicine. Cell, 148(6):1110-1122, 2012.
[Chi95] David Maxwell Chickering. A Transformational Characterization of Equivalent Bayesian Network Structures. In Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, UAI'95, page 87-98, San Francisco, CA, USA, 1995. Morgan Kaufmann Publishers Inc.
[CS23] Davin Choo and Kirankumar Shiragur. Subset verification and search algorithms for causal dags. arXiv preprint arXiv:2301.03180, 2023.
[CSB22] Davin Choo, Kirankumar Shiragur, and Arnab Bhattacharyya. Verification and search algorithms for causal DAGs. Advances in Neural Information Processing Systems, 35, 2022.
[DL05] Eric Davidson and Michael Levin. Gene regulatory networks. Proceedings of the National Academy of Sciences, 102(14):4935-4935, 2005.
[ER60] Paul Erdős and Alfréd Rényi. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci, 5(1):17-60, 1960.
[ES07] Frederick Eberhardt and Richard Scheines. Interventions and Causal Inference. Philosophy of science, 74(5):981-995, 2007.
[FLNP00] Nir Friedman, Michal Linial, Iftach Nachman, and Dana Pe'er. Using bayesian networks to analyze expression data. Journal of computational biology, 7(3-4):601-620, 2000.
$\left[\mathrm{FMT}^{+}21\right]$  Chris J Frangieh, Johannes C Melms, Pratiksha I Thakore, Kathryn R Geiger-Schuller, Patricia Ho, Adrienne M Luoma, Brian Cleary, Livnat Jerby-Arnon, Shruti Malu, Michael S Cuoco, et al. Multimodal pooled perturb-cite-seq screens in patient models define mechanisms of cancer immune evasion. Nature genetics, 53(3):332-341, 2021.
[GHD20] Juan L Gamella and Christina Heinze-Deml. Active invariant causal prediction: Experiment selection through stability. Advances in Neural Information Processing Systems, 33:15464-15475, 2020.
[WBL21] Marcel Wienöbst, Max Bannach, and Maciej Liskiewicz. Extendability of causal graphical models: Algorithms and computational complexity. In Cassio de Campos and Marloes H. Maathuis, editors, Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence, volume 161 of Proceedings of Machine Learning Research, pages 1248–1257. PMLR, 27–30 Jul 2021.
[YKU18] Karren Yang, Abigail Katcoff, and Caroline Uhler. Characterizing and learning equivalence classes of causal dags under interventions. In International Conference on Machine Learning, pages 5541-5550. PMLR, 2018.
[ZSU21] Jiaqi Zhang, Chandler Squires, and Caroline Uhler. Matching a desired causal state via shift interventions. Advances in Neural Information Processing Systems, 34:19923-19934, 2021.
A Meek Rules 15
B Preliminaries and Other Useful Results 15
C Another Example of Algorithm 1 15
D Remaining Proofs for Meek Separator 16
D.1 Proof for Lemma 11 16
D.2 Proof for Lemma 12 17
D.3 Proof for Lemma 13 17
D.4 Proof for Lemma 10 17
D.5 Proof for Lemma 14 18
D.6 Proof for Theorem 5 19
E Remaining Proofs for Subset Search 20
E.1 Lower Bound for Subset Search (Lemma 6) 20
E.2 Upper Bound for Subset Search (Theorem 7) 20
F Remaining Proofs for Causal Mean Matching 21
F.1 Proof for Lemma 8 22
F.2 Proof for Theorem 9 22
G Details of Numerical Experiments 23
G.1 Subset Search 23
G.2 Causal Mean Matching 24
A Meek Rules
Meek rules refer to a collection of four edge orientation rules that are proven to be sound and complete when applied to a set of arcs that possesses a consistent extension to a directed acyclic graph (DAG) [Mee95]. With the presence of edge orientation information, it is possible to iteratively apply Meek rules until reaching a fixed point, thereby maximizing the number of oriented arcs.
Definition 15 (Consistent extension). For a given graph  $G$ , a set of arcs is considered to have a consistent DAG extension  $\pi$  if there exists a permutation of the vertices satisfying the following conditions: (i) for every edge  $\{u, v\}$  in  $G$ , it is oriented as  $u \to v$  whenever  $\pi(u) < \pi(v)$ , (ii) there are no directed cycles, and (iii) all the given arcs are included in the extension.
Definition 16 (The four Meek rules [Mee95], see Figure 6 for an illustration).
R1 Edge  $\{a,b\} \in E\setminus A$  is oriented as  $a\to b$  if  $\exists c\in V$  such that  $c\rightarrow a$  and  $c\not\sim b$
R2 Edge  $\{a,b\} \in E\setminus A$  is oriented as  $a\to b$  if  $\exists c\in V$  such that  $a\rightarrow c\rightarrow b$
R3 Edge  $\{a,b\} \in E\setminus A$  is oriented as  $a\to b$  if  $\exists c,d\in V$  such that  $d\sim a\sim c,d\rightarrow b\gets c$  , and  $c\not\sim d.$
R4 Edge  $\{a,b\} \in E\setminus A$  is oriented as  $a\to b$  if  $\exists c,d\in V$  such that  $d\sim a\sim c,d\rightarrow c\rightarrow b$  , and  $b\not\sim d.$
Figure 6: An illustration of the four Meek rules
An algorithm [WBL21, Algorithm 2] has been developed to compute the closure under Meek rules efficiently. The algorithm runs in  $\mathcal{O}(d\cdot |E|)$  time, where  $d$  represents the degeneracy of the graph skeleton<sup>14</sup>.
B Preliminaries and Other Useful Results
Here we state some useful notation and results. For an arc  $u \to v$  in  $\mathcal{G}$ , we define  $R_1^{-1}(\mathcal{G}, u \to v) \subseteq V$  and  $R_k^{-1}(\mathcal{G}, u \to v) \subseteq 2^V$  to refer to interventions orienting an arc  $u \to v$ . Equivalently,  $R_1^{-1}(\mathcal{G}, u \to v) = \{w \in V : u \to v \in R(\mathcal{G}, w)\}$  and  $R_k^{-1}(\mathcal{G}, u \to v) = \{I \subseteq V : |I| \leq k, u \to v \in R(\mathcal{G}, I)\}$ .
Proposition 17 (Theorem 7 in [CS23]). Consider a DAG  $\mathcal{G} = (V,E)$  and intervention sets  $\mathcal{I},\mathcal{J}\subseteq$ $2^{V}$ . The following statements are true: 1)  $\operatorname {skel}(\mathcal{G}^{\mathcal{I}})$  is exactly the chain components of  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$ . 2)  $\mathcal{G}^{\mathcal{I}}$  does not have new  $\nu$ -structures. 3) For any two vertices  $u$  and  $v$  in the same chain component of  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$ , we have  $P_{\mathcal{A}_{\mathcal{G}},\mathcal{I}}(u) = P_{\mathcal{A}_{\mathcal{G}},\mathcal{I}}(v)$ . 4) If  $u\to v\in R(\mathcal{G},\mathcal{I})$ , then  $u$  and  $v$  belong to different chain components of  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$ . 5) Any acyclic completion of  $\mathcal{E}(\mathcal{G}^{\mathcal{I}})$  can be combined with  $R(\mathcal{G},\mathcal{I})$  to obtain a valid DAG that has the same essential graph and  $\mathcal{I}$ -essential graph as  $\mathcal{E}(\mathcal{G})$  and  $\mathcal{E}_{\mathcal{I}}(\mathcal{G})$  respectively. 6)  $R(\mathcal{G}^{\mathcal{I}},\mathcal{I}) = R(\mathcal{G},\mathcal{I})\setminus R(\mathcal{G},\mathcal{I})$ . 7)  $R(\mathcal{G},\mathcal{I}\cup \mathcal{I}) = R(\mathcal{G}^{\mathcal{I}},\mathcal{I})\dot{\cup} R(\mathcal{G},\mathcal{I})$ . 8)  $R(\mathcal{G},\mathcal{I}\cup \mathcal{I}) = R(\mathcal{G}^{\mathcal{I}},\mathcal{I})\dot{\cup} R(\mathcal{G}^{\mathcal{I}},\mathcal{I})\dot{\cup}\left(R(\mathcal{G},\mathcal{I})\cap R(\mathcal{G},\mathcal{I})\right)$ .
Lemma 18 (Theorem 10 in [CS23]). Let  $\mathcal{G} = (V,E)$  be a DAG without  $\nu$ -structures and  $u\to v$  in  $\mathcal{G}$  be unoriented in  $\mathcal{E}(\mathcal{G})$ . Then,  $R_1^{-1}(\mathcal{G},u\to v) = Des[w]\cap \mathsf{Anc}[v]$  for some  $w\in \mathsf{Anc}[u]$ .
C Another Example of Algorithm 1
We provide another example of Algorithm 1 in an incomplete graph, highlighting that Meek separator by solely focusing on the  $1/2$ -clique separator.
<sup>14</sup>A  $d$ -degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most  $d$ . Note that the degeneracy of a graph is typically smaller than the maximum degree of the graph.
(a) Ground-truth DAG.
(b) Iteration 0:  $K_{0} = \{1,2,3,4\}$ .
(d) Iteration 2:  $u_{1} = 4$ ,  $K_{2} = \{2,3\}$ .
(c) Iteration 1:  $u_0 = 1$ ,  $K_{1} = \{2,3,4\}$ .
(e) Iteration 3:  $u_{2} = 2$
Figure 7: An example of Algorithm 1 finding the Meek separator in an incomplete graph. The sets  $K_{i}$  are highlighted; oriented edges in  $\mathcal{E}_{u_i}(\mathcal{G})$  by intervening on  $u_{i}$  are in grey; connected components in  $CC(\mathcal{E}_{u_i}(\mathcal{G}))$  are in black. (a) ground-truth DAG  $\mathcal{G}$ . (b) suppose we take  $K_{0} = K = V(\mathcal{G})$  as the  $1/2$ -clique separator. (c) suppose we pick  $u_{0} = 1$  from  $K_{0}$ , then  $K_{1} = K_{0} \cap Q_{u_{0}} = \{2,3,4\}$ . (d) suppose we pick  $u_{1} = 4$  from  $K_{1}$ , then  $K_{2} = K_{1} \cap P_{u_{1}} = \{2,3\}$ . (e) suppose we pick  $u_{2} = 2$  from  $K_{2}$ , then Algorithm 1 terminates (line 7) returning meek separator  $\mathcal{I} = \{2\}$  and  $\mathcal{J} = \{1,4\}$  that helps find it.
D Remaining Proofs for Meek Separator
Here we provide all the remaining proofs for the Meek separator algorithm.
D.1 Proof for Lemma 11
Proof. Performing an intervention on node  $v$  results in orienting all its adjacent edges. Consequently, one of the connected components in  $CC(\mathcal{E}_v(\mathcal{G}))$  is  $\{v\}$ , and it is adequate to focus on the remaining connected components.
Suppose, for the sake of contradiction, that there exists a connected component  $\mathcal{H} \in CC(\mathcal{E}_v(\mathcal{G}))$  such that  $\mathcal{H}$  contains two vertices  $a$  and  $b$ , such that,  $a \in A_v$  and  $b \in B_v$ . Since  $a$  and  $b$  belong to the same connected component, we consider the path within  $H$  that connects these vertices. Notably, this path includes two adjacent vertices  $c$  and  $d$ , where  $c \in A_v$  and  $d \in B_v$ , and the edge  $(c,d)$  remains unoriented.
However, according to Lemma 18, intervening on any vertex within the set  $\mathsf{Des}[w] \cap \mathsf{Anc}[d]$  for some fixed  $w \in \mathsf{Anc}[c]$  will orient edge  $(c,d)$ . As  $w \in \mathsf{Anc}[c]$ , we have  $\mathsf{Des}[c] \subseteq \mathsf{Des}[w]$  and therefore  $\mathsf{Des}[c] \cap \mathsf{Anc}[d] \subseteq \mathsf{Des}[w] \cap \mathsf{Anc}[d]$ . Consequently as  $v \in \mathsf{Des}[c] \cap \mathsf{Anc}[d]$ , we get that intervening on  $v$  orients the edge  $(c,d)$ .
Hence, it is impossible for vertices  $a \in A_v$  and  $b \in B_v$  to belong to the same connected component. Consequently, we can conclude the proof.
D.2 Proof for Lemma 12
Proof. Since  $\mathcal{H}$  is a connected subgraph of  $\mathcal{G}$  and  $V(\mathcal{H}) \cap V(K) = \emptyset$ , removing  $K$  from  $\mathcal{G}$  does not result in the deletion of any edges or vertices from the subgraph  $\mathcal{H}$ . Consequently, the vertices in  $\mathcal{H}$  will remain connected even after the removal of  $K$ , and they will all belong to the same connected component in the graph  $\mathcal{G} \backslash K$ . Considering that  $K$  is an  $\alpha$ -separator, this further implies that  $|V(\mathcal{H})| \leq \alpha \cdot |V(\mathcal{G})|$ . Thus, we can conclude the proof.
D.3 Proof for Lemma 13
Proof. Since  $v_{i}$  precedes  $v_{i+1}$  in the true ordering defined by the underlying ground truth DAG  $\mathcal{G}$ , it follows that  $v_{i+1} \in \operatorname{Des}(v_{i})$ . Consequently, we can deduce that  $\operatorname{Des}(v_{i+1}) \subseteq \operatorname{Des}(v_{i})$ , which implies  $|B_{v_{i+1}}| \leq |B_{v_{i}}|$ . Additionally, note that  $|A_{v}| + |B_{v}| + 1 = |V(\mathcal{G})|$ . Therefore, it holds that  $|A_{v_{i+1}}| \geq |A_{v_{i}}|$ .
Furthermore, as  $v_{1}$  is the source node of the clique, removing  $K$  ensures that all vertices in  $A_{v_1}$  are still in the remaining graph. They also induce a connected subgraph. Otherwise suppose  $a\in A_{v_1}$  and  $b\in A_{v_1}$  are not connected. Then the parent  $c$  of  $v_{1}$  on a path from  $a$  to  $v_{1}$  and the parent  $d$  of  $v_{1}$  on a path from  $b$  to  $v_{1}$  are not connected. This creates a v-structure  $c\to v_1\gets d$ , contradicting  $\mathcal{G}$  being moral. Thus  $A_{v_1}$  is a connected subgraph with no intersection with  $K$ . As  $K$  is a  $\alpha$ -clique separator, we have by Lemma 12 that  $|A_{v_1}|\leq \alpha \cdot |V(\mathcal{G})|$ . We conclude the proof.
D.4 Proof for Lemma 10
Lemma 10 (Meek separator). Let  $\mathcal{G}$  be a moral DAG and  $K$  be a  $1/2$ -clique separator of  $\mathcal{G}$ . There exists a vertex  $u \in K$  satisfying the constraints  $|A_u| \leq |V(\mathcal{G})|/2$  and  $|A_v| > |V(\mathcal{G})|/2$  for all  $v \in Des(u) \cap K$ . Furthermore, such a vertex  $u$  satisfies one of the two conditions: 1). either  $u$  is a sink vertex<sup>15</sup> of  $\mathcal{G}[K]$ , or 2). there exists a vertex  $x$  such that,  $x \in Des(u) \cap K$  and  $\left((V(\mathcal{G}) \backslash Des[x]) \cap Des(u)\right) \cap K = \emptyset$  (i.e.,  $x$  and  $u$  are consecutive vertices in the valid permutation of clique  $K$ ). In both the cases respectively, either  $\{\{u\}\}$  or  $\{\{u\}, \{x\}\}$  is a  $1/2$ -Meek separator.
Proof. Consider the vertices  $v_{1}, \ldots, v_{k}$  of the clique  $K$  in the true ordering defined by the underlying ground truth  $\mathcal{G}$ . Since  $K$  is a  $1/2$ -clique separator, according to Lemma 13, we deduce that  $|A_{v_{1}}| \leq |V(\mathcal{G})| / 2$ . Let  $u$  denote the last vertex, in terms of the true ordering, within the clique  $K$  that satisfies  $A_{u} \leq |V(\mathcal{G})| / 2$ . It is important to note that  $u$  fulfills the constraints specified in the lemma.
Therefore there exists a vertex  $u$  that fulfills these constraints. Now we show that any vertex that fulfills these constraints will satisfy one of two conditions, and that either  $\{\{u\}\}$  or  $\{\{u\}, \{x\}\}$  is a  $1/2$ -Meek separator.
Suppose  $u$  is a sink vertex of  $K$ . Let  $\mathcal{H} \in CC(\mathcal{E}_u(\mathcal{G}))$  and note that according to Lemma 11,  $V(\mathcal{H}) = \{u\}$  or  $V(\mathcal{H}) \subseteq \operatorname{Des}(u)$  or  $V(\mathcal{H}) \subseteq V(\mathcal{G}) \backslash \operatorname{Des}[u]$ . Since  $|A_u| \leq |V(\mathcal{G})| / 2$ , any  $\mathcal{H}$  such that  $V(\mathcal{H}) \subseteq V(\mathcal{G}) \backslash \operatorname{Des}[u]$  satisfies  $|V(\mathcal{H})| \leq |V(\mathcal{G})| / 2$ . Now, suppose  $V(\mathcal{H}) \subseteq \operatorname{Des}(u)$ . In this case, we observe that  $V(\mathcal{H}) \cap K = \emptyset$  and  $\mathcal{H}$  is a connected subgraph of  $G$ . By utilizing Lemma 12, we immediately have  $|V(\mathcal{H})| \leq |V(\mathcal{G})| / 2$ . Therefore,  $\{\{u\}\}$  will be a 1/2-Meek separator.
Suppose  $u$  is not a sink vertex. Consider the vertex  $x$  that follows  $u$  within the clique  $K$  in the ordering defined by the DAG  $\mathcal{G}$ . Note that  $x \in \operatorname{Des}(u) \cap K$  and  $(\operatorname{Des}(u) \cap (V(\mathcal{G}) \backslash \operatorname{Des}[x])) \cap K = \varnothing$ . Furthermore from the definition of  $u$ , we also have that  $|A_x| > |V(\mathcal{G})| / 2$ , which further implies  $|B_x| \leq |V(\mathcal{G})| / 2$ . As intervening on more vertices only creates finer-grained connected components, we have that for any  $\mathcal{J} \subseteq V(\mathcal{G})$ ,  $w \in V(\mathcal{G})$ , and  $\mathcal{H}' \in CC(\mathcal{E}_{\mathcal{J} \cup \{w\}}(\mathcal{G}))$ , there exists an  $\mathcal{H}'' \in CC(\mathcal{E}_{\mathcal{J}}(\mathcal{G}))$  such that  $\mathcal{H}'$  is a subgraph of  $\mathcal{H}''$ .
${}^{15}u$  is a sink vertex if and only if it has no children.
Let  $\mathcal{I} = \{x,u\}$ , consider any  $\mathcal{H} \in CC(\mathcal{E}_{\mathcal{I}}(G))$  and note that there exist connected components  $M \in CC(\mathcal{E}_u(\mathcal{G}))$  and  $L \in CC(\mathcal{E}_x(\mathcal{G}))$  such that  $\mathcal{H}$  is a subgraph of both  $M$  and  $L$ . We apply Lemma 11 to deduce that  $V(M) = u$  or  $V(M) \subseteq \operatorname{Des}(u)$  or  $V(M) \subseteq V(\mathcal{G}) \setminus \operatorname{Des}[u]$  and  $V(L) = x$  or  $V(L) \subseteq \operatorname{Des}(x)$  or  $V(L) \subseteq V(\mathcal{G}) \setminus \operatorname{Des}[x]$ .
If  $V(M) = u$  or  $V(M) \subseteq V(\mathcal{G}) \backslash \operatorname{Des}[u]$  or  $V(L) = x$  or  $V(L) \subseteq \operatorname{Des}(x)$ , then  $V(\mathcal{H}) = u$  or  $V(\mathcal{H}) = x$  or  $V(\mathcal{H}) \subseteq A_u$  or  $V(\mathcal{H}) \subseteq B_x$ . As  $|A_u|, |B_x| \leq |V(\mathcal{G})| / 2$ , in all of these cases, we have that  $|V(\mathcal{H})| \leq |V(\mathcal{G})| / 2$ . Now consider the remaining cases where  $V(M) \subseteq \operatorname{Des}(u)$  and  $V(L) \subseteq V(\mathcal{G}) \backslash \operatorname{Des}[x]$ . As  $\mathcal{H}$  is a subgraph of both  $M$  and  $L$ , we have that  $\mathcal{H}$  is a subgraph of  $C$ , where  $C$  satisfies  $V(C) = \operatorname{Des}(u) \cap (V(\mathcal{G}) \backslash \operatorname{Des}[x])$ . As  $(\operatorname{Des}(u) \cap (V(\mathcal{G}) \backslash \operatorname{Des}[x])) \cap K = \emptyset$ , we have that  $V(C) \cap K = \emptyset$ . Furthermore, as  $\mathcal{H}$  is a subgraph of  $C$  we also have that  $V(\mathcal{H}) \cap K = \emptyset$ . As  $\mathcal{H}$  is a connected subgraph of  $G$  and  $V(\mathcal{H}) \cap K = \emptyset$ , by utilizing Lemma 12, we have that  $|V(\mathcal{H})| \leq |V(\mathcal{G})| / 2$ . Therefore in all cases, we have that  $|V(\mathcal{H})| \leq |V(\mathcal{G})| / 2$ , and thus  $\{\{u\}, \{x\}\}$  is a 1/2-Meek separator, which concludes the proof.
D.5 Proof for Lemma 14
Here we present a proof for Lemma 14, which describes the structure of the solution returned by our Meek separator algorithm. In order to prove this lemma, we first establish the following result.
Lemma 19. Let  $\mathcal{G}$  be a connected moral DAG and  $v\in V(\mathcal{G})$  be an arbitrary vertex. For any connected component  $\mathcal{H}\in CC(\mathcal{E}_v(\mathcal{G}))$ , there exists a directed path from  $v$  to  $\mathcal{H}$  in  $\mathcal{E}_v(\mathcal{G})$  if and only if  $\mathcal{H}$  satisfies  $V(\mathcal{H})\subseteq Des(v)$ . Furthermore, we can certify if  $\mathcal{H}$  is a subset of  $\mathcal{G}\backslash Des[v]$  or  $Des(v)$  in polynomial time.
Proof. Suppose there exists a directed path from  $v$  to a connected component  $\mathcal{H}$ , then note that all the vertices in this directed path are descendants of  $v$  and there exists a vertex  $u \in V(\mathcal{H})$  such that,  $u \in \operatorname{Des}(v)$ . Furthermore, by Lemma 11, we know that all the connected components  $\mathcal{H}$  should satisfy:  $V(\mathcal{H}) = \{v\}$  or  $V(\mathcal{H}) = \operatorname{Des}(v)$  or  $V(\mathcal{H}) = V(\mathcal{G}) \backslash \operatorname{Des}(v)$ . Using the fact that there exists a vertex  $u \in V(\mathcal{H})$  which belongs to  $\operatorname{Des}(v)$ . Therefore,  $\operatorname{Des}(v) \cap V(\mathcal{H})$  is not empty and we conclude that  $V(\mathcal{H}) \subseteq \operatorname{Des}(v)$ .
In the subsequent part of the proof, we examine the converse direction. Assume there exists a connected component  $\mathcal{H}$  such that  $V(\mathcal{H}) \subseteq \operatorname{Des}(v)$ , and we aim to demonstrate the existence of a directed path from  $v$  to  $\mathcal{H}$  in the interventional essential graph  $\mathcal{E}_v(\mathcal{G})$ . Since  $\mathcal{H} \subseteq \operatorname{Des}(v)$ , we consider the shortest directed path from  $v$  to  $\mathcal{H}$  in  $\mathcal{G}$ . Let  $w$  be the endpoint of this path  $P$  within  $\mathcal{H}$ .
Suppose that all edges in  $P$  are oriented. In this case, we have already found a directed path from  $v$  to  $\mathcal{H}$  in  $\mathcal{E}_v(\mathcal{G})$ , and our objective is achieved. Hence, we focus on the scenario where some edges in  $P$  are unoriented. Let  $P: v = v_0 \to v_1 \to \dots \to v_\ell \to w$  denote the vertices along the path  $P$ , and let  $v_j \sim v_{j+1}$  represent the first unoriented edge encountered.
Since all edges incident to  $v$  are oriented, we have  $v_{j} \neq v$ . Now, consider the situation where  $v_{j-1}$  and  $v_{j+1}$  are not adjacent. In such a case, the Meek rule R1 would orient the edge  $v_{j} \rightarrow v_{j+1}$ . However, since  $v_{j} \rightarrow v_{j+1}$  remains unoriented, it implies that the edge  $v_{j-1} \rightarrow v_{j+1}$  must exist. However, this contradicts the fact that  $P$  is the shortest directed path. Therefore, it must be the case that  $P$  is either entirely oriented or entirely unoriented. As some edges in  $P$  are oriented, we conclude that path  $P$  is completely oriented, thereby establishing the presence of a directed path from  $v$  to  $\mathcal{H}$  in the interventional essential graph  $\mathcal{E}_{v}(\mathcal{G})$ .