SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Sina Akbari
EPFL, Switzerland
sina.akbari@epfl.ch
Fateme Jamshidi
fateme.jamshidi@epfl.ch
Ehsan Mokhtarian
ehsan.mokhtarian@epfl.ch
Matthew J. Vowels
University of Lausanne, Switzerland
matthew.vowels@unil.ch
Jalal Etesami
TUM, Germany
j.etesami@tum.de
Causal identification is at the core of the causal inference literature, where complete algorithms have been proposed to identify causal queries of interest. The validity of these algorithms hinges on the restrictive assumption of having access to a correctly specified causal structure. In this work, we study the setting where a probabilistic model of the causal structure is available. Specifically, the edges in a causal graph exist with uncertainties which may, for example, represent degree of belief from domain experts. Alternatively, the uncertainty about an edge may reflect the confidence of a particular statistical test. The question that naturally arises in this setting is: Given such a probabilistic graph and a specific causal effect of interest, what is the subgraph which has the highest plausibility and for which the causal effect is identifiable? We show that answering this question reduces to solving an NP-complete combinatorial optimization problem which we call the edge ID problem. We propose efficient algorithms to approximate this problem and evaluate them against both real-world networks and randomly generated graphs.
A large proportion of questions of interest in various fields, including but not limited to psychology, social sciences, behavioral sciences, medical research, epidemiology, economy, etc., are causal in nature [33, 24, 8]. In order to estimate causal effects, the gold standard is performing controlled interventions and experiments. Unfortunately, such experiments can be prohibitively expensive, unethical, or impractical [10, 12]. In contrast, non-experimental data are comparatively abundant, and no expensive interventions are required to generate such data. This has motivated the development of numerous techniques for understanding whether a causal query can be answered using observational data. Specifically, if a particular causal query is identifiable, it means it can be expressed as a function of the observational distribution and thus estimated from observational data (at least in principle).
A significant body of the causal inference literature is dedicated to the identification problem [23, 6, 29, 26, 24]. In particular, [13] presented a complete algorithmic approach to decide the identifiability of a specific query and proved that Pearl's do calculus is complete. Furthermore, [26] provided graphical criteria to decide the identifiability based on the hedge criterion. The problem of identifiability was later generalized to deal with interventional data [21, 20], selection bias [1], context-specific independence relations [31, 15] and causal bandits [16]. However, all of these results hinge on the full specification of the causal structure, i.e., access to a correctly specified Acyclic Directed Mixed Graph (ADMG) that models the causal dynamics of the system. This requirement is restrictive in a number of ways. (i) The causal identification problem is concerned with inference from the
observational data, but the ADMG cannot be inferred from the observational distribution alone. (ii) Structure learning methods rely heavily on statistical tests [27, 9, 5, 4, 17], which are prone to errors arising from lack of sufficient data and method-specific limitations [25]. This can in turn result in misspecification of the causal structure.
A notable work to address the second challenge is [22], where the authors proposed a causal discovery method that assigns a probability to the existence of each edge, as opposed to making "hard" decisions based on a single or limited number of tests. This is achieved by updating the posterior probability of a direct link between two variables, denoted as  $X$  and  $Y$ , using conditional dependence tests that incorporate independent evidence. The authors demonstrate that the posterior probability denoted as  $p_i$ , of a link between  $X$  and  $Y$  after performing  $i$  dependence tests ( $d_j$ , where  $j$  ranges from 1 to  $i$ ) is given by the following equation:
$$
p _ {i} = \frac {p _ {i - 1} d _ {i}}{p _ {i - 1} d _ {i} + (1 - p _ {i - 1}) (G + 1 - d _ {i})},
$$
where  $G$  represents a factor ranging from 0 to 1, and it can be interpreted as the measure of "importance" or relevance associated with the truthfulness of each test, denoted as  $d_{i}$ . Other closely linked work include [7] and [4], where an optimal transport (OT)-based causal discovery approach is proposed that assigns a score to each edge of the graph denoting the strength of that edge.
While previous research, such as [22], has focused on probabilistic learning of causal graphs, the problem of causal effect identification on these graphs remains unexplored. This paper introduces a novel approach to address causal effect identification in a scenario where only a probabilistic model of the causal structure is available, as opposed to a complete specification of the structure itself. For instance, an ADMG  $\mathcal{G}$  is given along with probabilities assigned to each edge of  $\mathcal{G}$ . An example is shown in Figure 1a. These probabilities could represent uncertainties arising from statistical tests or the strength of belief of domain experts concerning the plausibility of the existence of an edge. Under this setting, each ADMG on the set of vertices of  $\mathcal{G}$  is assigned its own plausibility score. Since the causal structure is not deterministic anymore, answering questions such as "is the causal effect  $P(Y|do(X))$  identifiable?" also becomes probabilistic in nature. One can compare the overall plausibility of different subgraphs in which the causal effect is identifiable and then select the graph which maximizes the plausibility. In this work, for a specific causal query  $P(Y|do(X))$ , we first answer the question, "Which graph has the highest plausibility among those compliant with the probabilistic ADMG model that renders  $P(Y|do(X))$  identifiable?" The answer to this question allows us to quantify how confident we are about the causal identification task given the combination of the data at hand and the corresponding probabilistic model.
As the causal identification task is carried out through an identification formula that is based on the causal structure, our second focus is on deriving an identification formula for a given causal query that holds with the highest probability. This problem poses a greater challenge compared to the former, as a single identification formula can be valid for various graphs. Consequently, the probability that a particular identification formula is valid for a causal query becomes an aggregate probability across all graphs where this formula holds. We shall illustrate this point in more detail through Example 1 in Section 2. The intricacy of characterizing all valid graphs for a given formula makes it an intractable and open problem. To surmount this obstacle, we establish that if an identification formula is valid for a causal graph, it remains valid for all edge-induced subgraphs of that graph. This allows us to form a surrogate problem (see Problem 2 in Section 2.1) that recovers a causal graph with the highest aggregated probability of its subgraphs. The probability of the resulting graph provides a lower bound on the highest probable identification formula.
Both problems discussed in this work are aimed at evaluating the plausibility of performing causal identification for a specific query given a dataset and a non-deterministic model describing the causal structure. To sum up, our main contributions are as follows.
1. We study the problem of causal identifiability in probabilistic causal models, where there are uncertainties about the existence of edges and whether a given causal effect is identifiable. More precisely, we consider two problems: 1) finding the most probable graph that renders a desired causal query identifiable, and 2) finding the graph with the highest aggregate probability over its edge-induced subgraphs that renders a desired causal query identifiable.
(a) ADMG  $\mathcal{G}$ .
(b)  $\mathcal{G}_1\subseteq \mathcal{G}$
Figure 1: (a) An example of a probabilistic ADMG  $\mathcal{G}$  with corresponding edge probabilities. (b) and (c) are two different subgraphs of  $\mathcal{G}$  in which  $Q[y]$  is identifiable.
(c)  $\mathcal{G}_2\subseteq \mathcal{G}$
2. We show that both aforementioned problems reduce to a special combinatorial optimization problem which we call the edge ID problem. Moreover, we prove that the edge ID problem is NP-complete, and thus, so are both of the problems.
3. We propose several exact and heuristic algorithms for the aforementioned problems and evaluate their performances through different experiments.
We utilize small letters for variables and capital letters for sets of variables. Calligraphic letters are used to denote graphs. An acyclic directed mixed graph (ADMG)  $\mathcal{G} = (V^{\mathcal{G}}, E_d^{\mathcal{G}}, E_b^{\mathcal{G}})$  is defined as an acyclic graph on the vertices  $V^{\mathcal{G}}$ , where  $E_d^{\mathcal{G}} \subseteq V^{\mathcal{G}} \times V^{\mathcal{G}}$  and  $E_b^{\mathcal{G}} \subseteq \binom{V^{\mathcal{G}}}{2}$  are the set of directed and bidirected edges among the vertices, respectively. With slight abuse of notation, if  $e \in E_d^{\mathcal{G}} \cup E_b^{\mathcal{G}}$ , we write  $e \in \mathcal{G}$ . We use  $\mathcal{G}' \subseteq \mathcal{G}$  when  $\mathcal{G}'$  is an edge-induced subgraph of  $\mathcal{G}$ , i.e.,  $\mathcal{G}' = (V^{\mathcal{G}'}', E_d^{\mathcal{G}'}', E_b^{\mathcal{G}'}')$ , where  $V^{\mathcal{G}'} = V^{\mathcal{G}}$  and  $E_i^{\mathcal{G}'} \subseteq E_i^{\mathcal{G}}$  for  $i \in \{b, d\}$ . We denote by  $\mathcal{G}[X]$  the vertex-induced subgraph of  $\mathcal{G}$  over the subset of vertices  $X \subseteq V^{\mathcal{G}}$ . For a set of vertices  $X$ , we denote by  $Anc_{\mathcal{G}}(X)$  the set of vertices in  $\mathcal{G}$  that have a directed path to  $X$ . Note that  $X \subseteq Anc_{\mathcal{G}}(X)$ . Let  $P_X(Y)$  be a shorthand for  $P(Y|do(X))$ , and  $P^M(\cdot)$  denote the distribution of variables described by the causal model  $M$ .
Definition 2.1 (Identifiability [24]). Given a causal ADMG  $\mathcal{G} = (V^{\mathcal{G}},E_d^{\mathcal{G}},E_b^{\mathcal{G}})$ , and two disjoint subsets of variables  $X,Y\subseteq V^{\mathcal{G}}$ , the causal effect of  $X$  on  $Y$ , denoted by  $P_{X}(Y)$ , is identifiable in  $\mathcal{G}$  if  $P_X^{M_1}(Y) = P_X^{M_2}(Y)$  for any two models  $M_{1}$  and  $M_2$  that induce  $\mathcal{G}$  and  $P^{M_1}(V^{\mathcal{G}}) = P^{M_2}(V^{\mathcal{G}}) > 0$ .
Definition 2.2 (Valid identification formula). For a causal ADMG  $\mathcal{G}$  over variables  $V^{\mathcal{G}}$  and a causal query  $P_{X}(Y)$ , we say a functional  $\mathcal{F}$  defined on the probability space over  $V^{\mathcal{G}}$  is a valid identification formula for  $P_{X}(Y)$  in  $\mathcal{G}$  if  $P_{X}^{M_{1}}(Y) = P_{X}^{M_{2}}(Y) = \mathcal{F}(P^{M_{1}}(V^{\mathcal{G}})) = \mathcal{F}(P^{M_{2}}(V^{\mathcal{G}}))$  for any two models  $M_{1}$  and  $M_{2}$  that induce  $\mathcal{G}$  and  $P^{M_1}(V^{\mathcal{G}}) = P^{M_2}(V^{\mathcal{G}}) > 0$ .
For any query  $P_{X}(Y)$ , let  $[\mathcal{G}]_{Id(P_X(Y))}$  denote the set of subgraphs of  $\mathcal{G}$  in which  $P_{X}(Y)$  is identifiable (note that if  $\mathcal{G}$  is complete graph,  $[\mathcal{G}]_{Id(P_X(Y))}$  is the set of all graphs in which  $P_{X}(Y)$  is identifiable.) We denote by  $Q[Y]$  the causal effect of  $V\backslash Y$  on  $Y$ , i.e.,  $Q[Y] = P(Y|do(V\backslash Y))$ .
Definition 2.3 (District). For ADMG  $\mathcal{G} = (V^{\mathcal{G}},E_d^{\mathcal{G}},E_b^{\mathcal{G}})$ , let  $\mathcal{G}_{\leftrightarrow}$  denote the edge-induced subgraph of  $\mathcal{G}$  over its bidirected edges.  $X\subseteq V^{\mathcal{G}}$  is a district (aka c-component) in  $\mathcal{G}$  if  $\mathcal{G}_{\leftrightarrow}[X]$  is connected.
Definition 2.4 (Hedge [26]). Let  $\mathcal{G}$  be an ADMG, and  $Y \subsetneq X$  be two subsets of its vertices, where  $Y$  is a district in  $\mathcal{G}[Y]$ . Vertices  $X$  form a hedge for  $Q[Y]$  if  $X$  is a district in  $\mathcal{G}[X]$  and  $\operatorname{Anc}_{\mathcal{G}[X]}(Y) = X^1$ .
Definition 2.5 (Maximal hedge [2]). For ADMG  $\mathcal{G}$  with set vertices  $Y$ , let  $X$  be the union of all hedges formed for  $Q[Y]$ . Graph  $\mathcal{G}[X]$ , denoted by  $\mathbf{M} \mathbf{H}(\mathcal{G}, Y)$ , is called the maximal hedge for  $Q[Y]$ .
As an example, both sets  $\{t,x\}$  and  $\{z,x\}$  form a hedge for  $Q[x]$  in  $\mathcal{G}$  in Figure 1a, and  $\mathcal{G}[\{x,z,t\} ]$  is the maximal hedge for  $Q[x]$ .
<sup>1</sup>We showed in our previous work [2] that this intuitive definition is equivalent to the standard definition of hedge in [26].
2.1 Problem setup
Let  $\mathcal{G} = (V^{\mathcal{G}},E_d^{\mathcal{G}},E_b^{\mathcal{G}})$  be an ADMG, where  $V^{\mathcal{G}}$  is the set of vertices each representing an observed variable of the system,  $E_{d}^{\mathcal{G}}$  is the set of directed edges, and  $E_{b}^{\mathcal{G}}$  is the set of bidirected edges among  $V^{\mathcal{G}}$ . We know a priori that the true ADMG describing the system is an edge-induced subgraph of  $\mathcal{G}^2$ , and we are given a probability map that indicates for each subgraph of  $\mathcal{G}$  such as  $\mathcal{G}_s$ , with what probability  $\mathcal{G}_s$  is the true causal ADMG of the system. We denote this probability as  $P(\mathcal{G}_s)$ . For instance, if edge probabilities  $p_e$  are assumed to be mutually independent,  $P(\mathcal{G}_s)$  takes the form:
$$
P \left(\mathcal {G} _ {s}\right) = \prod_ {e \in \mathcal {G} _ {s}} p _ {e} \prod_ {e \notin \mathcal {G} _ {s}} \left(1 - p _ {e}\right). \tag {1}
$$
In what follows, we will refer to  $P(\mathcal{G}_s)$  simply as the probability of the ADMG  $\mathcal{G}_s$ . The first problem of our interest is formally defined as follows.
Problem 1. We consider the problem of finding the most probable edge-induced subgraph of  $\mathcal{G}$ , in which the causal effect  $Q[Y]^3$  is identifiable. That is, the goal is to find the ADMG  $\mathcal{G}^*$  defined by
$$
\mathcal {G} ^ {*} := \underset {\mathcal {G} _ {s} \subseteq \mathcal {G}, \mathcal {G} _ {s} \in [ \mathcal {G} ] _ {I d (Q [ Y ])}} {\arg \max } P (\mathcal {G} _ {s}). \tag {2}
$$
We will prove in Lemma 3 that if  $Q[Y]$  is identifiable in  $\mathcal{G}$ , then it is also identifiable in every edge-induced subgraph of  $\mathcal{G}$ . In other words, if  $\mathcal{G}$  is a feasible solution to the above optimization problem, so are all its edge-induced subgraphs. Furthermore, the same identification functional that is valid w.r.t.  $\mathcal{G}$ , is also valid w.r.t. every subgraph of  $\mathcal{G}$ . Let us illustrate this first on an example.
Example 1. Consider the ADMG in Figure 1a. With the given edge probabilities and assuming independence among the edge probabilities, the subgraph of  $\mathcal{G}$  illustrated in Figure 1b has probability  $0.7 \times 0.7 \times 0.1 = 0.049$ , whereas the subgraph of Figure 1c has probability  $0.3 \times 0.3 \times 0.9 = 0.081$  (see Eq. (1)). If we were to solve Problem 1, we would choose  $\mathcal{G}_2$  over  $\mathcal{G}_1$ , as it has a higher probability. Now consider identification formulas in  $\mathcal{G}_1$  and  $\mathcal{G}_2$ , respectively:
$$
\mathcal {F} _ {1}: Q [ Y ] = P (Y | X), \qquad \mathcal {F} _ {2}: Q [ Y ] = \sum_ {Z, T} P (Y | X, Z, T) P (Z, T).
$$
$\mathcal{F}_1$  is a valid identification formula for any edge-induced subgraph of  $\mathcal{G}_1$  (see Lemma 3). Analogously,  $\mathcal{F}_2$  is valid for all edge-induced subgraphs of  $\mathcal{G}_2$ . If we consider the aggregate probability of the subgraphs of  $\mathcal{G}_1$  and  $\mathcal{G}_2$ , i.e.,
$$
\sum_ {\hat {\mathcal {G}} \subseteq \mathcal {G} _ {1}} P (\hat {\mathcal {G}}) = 1 - 0. 9 = 0. 1, \quad \text {v s .} \quad \sum_ {\hat {\mathcal {G}} \subseteq \mathcal {G} _ {2}} P (\hat {\mathcal {G}}) = (1 - 0. 7) \times (1 - 0. 7) = 0. 0 9,
$$
then we should prefer choosing  $\mathcal{G}_1$  over  $\mathcal{G}_2$ , as its identification formula  $\mathcal{F}_1$  is more likely to be valid than  $\mathcal{F}_2$  considering the fact that for all its subgraphs, the identification functional  $\mathcal{F}_1$  is still valid.
Plausibility of a certain identification functional  $\mathcal{F}$  is the sum of the probabilities of all graphs in which  $\mathcal{F}$  is valid given the query of interest. Finding the most plausible identification formula for a given query requires computing the plausibility of all formulae. Since the space of all formulae is intractable, alternatively, we can enumerate all valid formulae for a given graph which changes the problem's search space to the space of all graphs. However, this is yet another challenging and, to the best of our knowledge, an open problem. Instead, we proposed a surrogate problem that yields a lower bound for the most plausible identification formula and identifies a causal graph where this formula is valid. To do so, we use the result of Lemma 3 that shows when an identification functional is valid in a causal graph, it is also valid in all its edge-induced subgraphs.
Problem 2. Consider the problem of finding the edge-induced subgraph  $\mathcal{H}^*$  of  $\mathcal{G}$  with a maximum aggregate probability of its subgraphs, in which  $Q[Y]$  is identifiable. Formally,
$$
\mathcal {H} ^ {*} := \underset {\mathcal {G} _ {s} \subseteq \mathcal {G}, \mathcal {G} _ {s} \in [ \mathcal {G} ] _ {I d (Q [ Y ])}} {\arg \max } \sum_ {\hat {\mathcal {G}} \subseteq \mathcal {G} _ {s}} P (\hat {\mathcal {G}}). \tag {3}
$$
Note that in case such information is not available, one can simply take  $\mathcal{G}$  to be a complete graph over both its directed and bidirected edges, as every graph is a subgraph of the complete graph. Even when a certain  $\mathcal{G}$  is available but the investigator is uncertain about it, the complete graph can serve as a conservative choice to eliminate uncertainty.
<sup>3</sup>As we shall discuss in Remark 2.1, our results are not limited to causal queries of the form  $Q[Y]$ . They are applicable to general causal queries of the form  $P_X(Y)$  as long as the set of ancestors of  $Y$  are known.
In other words, we are looking for a causal graph  $\mathcal{H}^*$  with the maximum aggregate probability of its subgraphs among the graphs in  $[\mathcal{G}]_{Id(Q[Y])}$ , i.e., the graphs in which  $Q[Y]$  is identifiable. Running an identification algorithm on  $\mathcal{H}^*$  yields an identification formula for  $Q[Y]$ , which is valid at least with the aggregate probability of the subgraphs of  $\mathcal{H}^*$ . Therefore, Problem 2 is a surrogate for recovering the identification formula with the highest plausibility. In the sequel, for simplicity, we study Problems 1 and 2 under the following assumption. However, as proved in Appendix C, our results are valid in a more general setting where we allow only for perfect negative or positive correlations among the edges. An example of a perfect negative correlation between two edges is that both cannot exist simultaneously. Appendix C.1 discusses the significance of this generalization.
Assumption 2.1. The edges of  $\mathcal{G}$  are mutually independent, i.e., the probability of a subgraph  $\mathcal{G}_s$  of  $\mathcal{G}$  is of the form (1).
Remark 2.1. It is noteworthy that our results are not limited to causal queries of the form  $Q[Y] = P(Y|do(V^{\mathcal{G}}\setminus Y))$ . They can be applied to general causal queries of the form  $P_{X}(Y)$  if the set  $\text{Anc}_{\mathcal{G}\backslash X}(Y)$  is known. This is because the causal query  $P_{X}(Y)$  can be expressed as  $\sum_{\text{Anc}_{\mathcal{G}\backslash X}(Y)\backslash Y}Q[\text{Anc}_{\mathcal{G}\backslash X}(Y)]$ , where  $\text{Anc}_{\mathcal{G}\backslash X}(Y)$  is the set of ancestors of  $Y$  in  $\mathcal{G}$  after removing the vertices of  $X$ . Furthermore,  $P_{X}(Y)$  is identifiable in  $\mathcal{G}$  if and only if  $Q[\text{Anc}_{\mathcal{G}\backslash X}(Y)]$  is identifiable in  $\mathcal{G}$  [30, 26, 14]. Note that the assumption that  $\text{Anc}_{\mathcal{G}\backslash X}(Y)$  is known is not equivalent to precluding uncertainty on the directed edges (as in the case of fixing the edge probabilities to 0 or 1), but it rather imposes a perfect correlation type of constraint. Consider for instance the three graphs of Figure 2, where all of them share the same set  $\text{Anc}_{\mathcal{G}\backslash X}(Y) = \{z,t\}$ . In fact, knowing this set forces a constraint of the type that if the edge  $z\to y$  does not exist, the path  $z\to t\to y$  must.
3 Reduction to edge ID problem and establishing complexity
We begin this section with the following lemma, to which we referred before. Thereafter, we discuss the hardness of the two problems considered in this work. For any causal query  $P_X(Y)$  and ADMG  $\mathcal{G}$ , if  $\mathcal{F}$  is a valid identification formula for  $P_X(Y)$  in  $\mathcal{G}$  (Def. 2.2), then  $\mathcal{F}$  is a valid identification formula for  $P_X(Y)$  in any  $\mathcal{G}' \subseteq \mathcal{G}$ . All proofs are presented in Appendix A. In what follows, we first formally define the edge ID problem and then show the equivalence of Problems 1 and 2 to the edge ID problem under Assumption 2.1.
Definition 3.1 (Edge ID problem). For ADMG  $\mathcal{G} = (V^{\mathcal{G}},E_d^{\mathcal{G}},E_b^{\mathcal{G}})$ , a set of non-negative edge weights  $W_{\mathcal{G}} = \{w_e\geq 0|e\in \mathcal{G}\}$ , and a causal query  $Q[Y]$  for  $Y\subseteq V^{\mathcal{G}}$ , the objective of the edge ID problem is to find the set of edges  $E^{*}\subseteq E_{d}^{\mathcal{G}}\cup E_{b}^{\mathcal{G}}$  with minimum aggregated weight (cost), such that  $Q[Y]$  is identifiable in the graph resulting from removing  $E^{*}$  from  $\mathcal{G}$ . Formally,
$$
E ^ {*} := \underset {E \subseteq E _ {d} ^ {\mathcal {G}} \cup E _ {b} ^ {\mathcal {G}}} {\arg \min } \sum_ {e \in E} w _ {e}, \quad \mathbf {s}. \mathbf {t}. \quad \mathcal {G} ^ {\prime} = \left(V ^ {\mathcal {G}}, E _ {d} ^ {\mathcal {G}} \backslash E, E _ {b} ^ {\mathcal {G}} \backslash E\right) \in [ \mathcal {G} ] _ {I d (Q [ Y ])}. \tag {4}
$$
That is, the cost of removing a set of edges from  $\mathcal{G}$  is the sum of the weights of each individual edge.
The following result unifies the two problems considered in this work by establishing their equivalence to the edge ID problem. This is done by transforming Problems 1 and 2 with multiplicative objectives into the edge ID problem that has an additive objective.
Lemma 3.2. Under Assumption 2.1, Problem 1 is equivalent to the edge ID problem with the edge weights chosen to be the log propensity ratios, i.e.,  $w_{e} = \max \{0,\log (\frac{p_{e}}{1 - p_{e}})\}$ ,  $\forall e\in \mathcal{G}$ . Moreover, Problem 2 is equivalent to the edge ID problem with the choice of weights  $w_{e} = -\log (1 - p_{e})$ $\forall e\in \mathcal{G}$ . That is, an instance of Problems 1 and 2 can be reduced to an instance of the edge ID problem in polynomial time, and vice versa.
As we mentioned earlier, the equivalence of these three problems can be established in more general settings than what is described under Assumption 2.1. We refer the interested reader to Appendix C for a discussion on one such setting. The following result shows no polynomial-time algorithm for solving these three problems exists unless  $\mathrm{P} = \mathrm{NP}$ .
Theorem 3.3. The edge ID problem is NP-complete.
Theorem 3.3 is established through a reduction from another NP-complete problem, namely the minimum-cost intervention problem (MCIP) [2, 3], which we shall discuss in Section 4.3. Theorem
Figure 2: Three different graphs that share the same set  $\text{Anc}_{\mathcal{G}}(\{y\}) = \{z, t\}$ .
3.3 is a key result which shows the hardness of recovering the most plausible graph in which a specified causal effect of interest is identifiable.