SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Lemma 3.2 states the equivalence of Problems 1 and 2 with the edge ID problem under Assumption 2.1. However, as mentioned in the main text, this equivalence holds in the more general setting where we allow for perfect negative correlations among edges. As an example, consider the graph of Figure 6. Suppose that the performed statistical independence tests show that the two variables  $z$  and  $y$  are independent of each other with high confidence. As a result, the expert believes that the edges  $(z, x)$  and  $(x, y)$  must not exist simultaneously, as otherwise the causal path from  $z$  to  $y$  would make them dependent. In such cases, the belief of the expert can be modeled as probabilities  $p$  and  $q$  assigned to the existence of the edges  $(z, x)$  and  $(x, y)$ , respectively, as well as a perfect negative correlation between them.
Note that the aforementioned constraint, i.e., that the edges do not exist simultaneously, can be specified for any number of edges, not limited to two edges only. For instance, the expert might believe at least one of the edges along a causal path of length  $n$  must not exist in the true ADMG describing the system. This belief can be modeled as an extra constraint in the optimization of Equations 2 and 3. We show that with the specification of such negative correlations, Problems 1 and 2 can still be cast as an instance of the edge ID problem. Therefore, the results presented in this work are valid in this setting as well.
Assumption C.1. The edges in  $\mathcal{G}$  are assigned probabilities  $p_e, \forall e \in \mathcal{G}$ , and perfect negative correlations are defined among subsets of edges. More precisely, for any subset  $E \subseteq E_d^{\mathcal{G}} \cup E_b^{\mathcal{G}}$ , there is either 1) no constraint (mutually independent), or 2) the constraint that at least one of the edges in  $E$  must not exist in the true ADMG (perfect negative correlation).
Proposition C.1. Under Assumption C.1, there exists a reduction from Problems 1 and 2 to the edge ID problem and vice versa with the time complexity in the order of  $\mathcal{O}(|C| \cdot |V^{\mathcal{G}}| + |E_d^{\mathcal{G}} \cup E_b^{\mathcal{G}}|)$ , where  $C$  is the set of perfect correlation constraints.
Proof. First note that we proved the equivalence of Problems 1 and 2 with the edge ID problem without the perfect correlation constraints in Lemma 3.2. As a result, under assumption C.1, i.e., by adding the perfect correlation constraints, Problems 1 and 2 are equivalent to a modified edge ID problem with those constraints. But we claim that there exists and instance of the original unconstrained edge ID problem which is equivalent to these problems. To see this, first note that we know from Corollary A.11 that the edge ID problem is equivalent to MCIP. Therefore, it suffices to show that there exists an instance of MCIP which is equivalent to the constrained edge ID mentioned above. To this end, consider the transform  $\mathcal{T}_2(\mathcal{G}, Y)$  introduced in Section A.2. This transformation
Figure 7: Integrating the perfect negative correlation constraint into MCIP.
maps an instance of the edge ID problem to an instance of MCIP. Applying this transformation to the constrained edge ID problem, we can map the constrained edge ID to an instance of MCIP with extra constraints, with transforming the constraints as well. That is, if for instance, there is a perfect negative correlation among the edges  $e_1, e_2$  in  $\mathcal{G}$ , this constraint is mapped to a negative perfect correlation on the corresponding vertices in  $\mathcal{H}$ , namely  $\mathcal{T}_2(e_1), \mathcal{T}_2(e_2)$ . In words, this constraint would be that at least one of  $\mathcal{T}_2(e_1)$  and  $\mathcal{T}_2(e_2)$  must be intervened upon. We show that such constraints can be integrated into the original definition of MCIP.
Suppose we have an MCIP problem in ADMG  $\mathcal{G}$  with query  $Q[Y]$ , with the extra constraint that at least one of the vertices  $X \subseteq V^{\mathcal{G}}$  must be intervened upon. Consider the example of  $X = \{x_1, x_2, x_3\}$  in Figure 7. We build a new ADMG  $\mathcal{G}'$  by adding vertices  $\{x'|x \in X\}$ , i.e., a new vertex corresponding to each vertex in  $X$ , along with an auxiliary vertex  $\hat{y}$  to  $\mathcal{G}$ . We fix a random ordering over the vertices of  $X$ , and denote the set of vertices of  $X$  as  $x_1, \ldots, x_m$ . We add the directed edges  $(x_i, x_i')$  to  $\mathcal{G}'$ , as well as the bidirected edges  $\{x_i, x_i'\}$ . Further, we draw directed edges  $(x_i', x_{i+1}')$  for every  $1 \leq i < m$ . Finally, we draw the directed edge  $(x_m', \hat{y})$  and the bidirected edge  $\{x_1, \hat{y}\}$ . Refer to the graph in Figure 7 for an example with  $X = \{x_1, x_2, x_3\}$ . Note that the set  $X \cup X' \cup \{\hat{y}\}$  forms a hedge for  $Q[\hat{y}]$ , where  $X' = \{x'|x \in X\}$ . Now it suffices to set the cost of intervention on vertices of  $X'$  to infinity, and consider MCIP for the query  $Q[Y \cup \{\hat{y}\}]$  in  $\mathcal{G}'$ . It is straightforward to see that the objective of this problem would be to find the minimum cost intervention for identification of  $Q[Y]$ , with the constraint that at least one of the vertices in  $X$  must be intervened on. Note that as soon as one vertex in  $X$  gets intervened upon, there is no hedge left for  $Q[\hat{y}]$ . Also it is noteworthy that adding this structure does not add any new hedges formed for  $Q[Y]$ , since the structure only includes new descendants for  $X$  which have no directed paths to  $Y$ . Also note that the vertices  $X'$  and  $\hat{y}$  are specific to the very constraint corresponding to the set of vertices  $X$ . For any constraint, we add such a structure to  $\mathcal{G}$ . The number of vertices (and therefore the time complexity) is at most in the order  $\mathcal{O}(|C| \cdot |V^{\mathcal{G}}|)$ , where  $C$  is the set of constraints.
C.1 Further applications
The relaxation provided in this Appendix allows the approaches proposed in this work to be applicable to a more general set of problems. Herein, we discuss one such application.
Let us assume we run our algorithm which returns the subgraph with the highest probability,  $\mathcal{G}_1$ . However, the probability that  $\mathcal{G}_1$  is the true causal structure describing the system might be too low. In such a case, the researcher might be interested in having a ranking of most probable graphs (for instance, the 10 most probable graphs), rather than only one most probable graph. This could be helpful for instance, when a unique identification formula is valid in a few of these graphs, or the researcher is interested in identifying more than one causal query. The methods discussed in this work along with the relaxation proposed in this appendix provide the tools to recover such a ranking (of the most probable graphs in which a query is identifiable). To see this, note that based on what we proposed in this Appendix, perfect negative correlation constraints can be added to the edge ID problem without additional computational cost. So we begin by solving the original problem, which yields a graph  $\mathcal{G}_1$ . We then solve it for a second time (i.e., re-run the algorithm), with the only difference that we add the perfect negative correlation constraint over the set of all edges of  $\mathcal{G}_1$  (i.e., we force the algorithm to exclude at least one of the edges of  $\mathcal{G}_1$ .) The solution to this problem (let us call it  $\mathcal{G}_2$ ) is the highest probability graph among all subgraphs except  $\mathcal{G}_1$ , i.e., it is the second highest
probability graph in which the query is identifiable. Continuing in this manner, running the algorithm  $n$  times would give us a ranking of the  $n$  highest probability graphs.
D Heuristic algorithms
Algorithm 2 was devised considering the fact that every hedge formed for  $Q[Y]$  must include a vertex that has a bidirected edge to  $Y$ . As mentioned in Section 4.2, an analogous approach, summarized in Algorithm 4, uses the fact that any hedge formed for  $Q[Y]$  must include a parent of  $Y$ .
Let  $Y \subseteq V^{\mathcal{G}}$  be a set of vertices of  $\mathcal{G}$  such that  $\mathcal{G}[Y]$  comprises of only one district. Let  $Z := \{z \in V^{\mathcal{G}} | \exists y \in Y : (z, y) \in E_d^{\mathcal{G}}\} \setminus Y$  denote the set of vertices that have at least one directed edge to a vertex in  $Y$ , i.e., the parents of  $Y$  excluding  $Y$ . Any hedge formed for  $Q[Y]$  contains at least one vertex of  $Z$ . As a result, in order to eliminate all the hedges formed for  $Q[Y]$ , it suffices to ensure that none of the vertices in  $Z$  appear in the final hedge. To this end, for any  $z \in Z$ , it suffices to either remove all the directed edges between  $z$  and  $Y$ , or eliminate all the bidirected paths from  $z$  to  $Y$ . The problem of eliminating all bidirected paths from  $Z$  to  $Y$  can be cast as a minimum cut problem between  $Z$  and  $Y$  in the edge-induced subgraph of  $\mathcal{G}$  over its bidirected edges. To add the possibility of removing the directed edges between  $Z$  and  $Y$ , we add an auxiliary vertex  $z^*$  to the graph and draw a bidirected edge between  $z^*$  and every  $z \in Z$  with weight  $w = \sum_{y \in Y} w_{(z, y)}$ , i.e., the sum of the weights of all directed edges between  $z$  and  $Y$ . Note that  $z$  can have directed edges to multiple vertices in  $Y$ . We then solve the minimum cut problem for  $z^*$  and  $Y$ . If an edge between  $z^*$  and  $z \in Z$  is in the solution to this min-cut problem, it translates to removing all the directed edges from  $z$  to  $Y$  in the original problem. Note that we can run the algorithm on the maximal hedge formed for  $Q[Y]$  in  $\mathcal{G}$  rather than  $\mathcal{G}$  itself.
Algorithm 4 Heuristic algorithm 2.
1: function HEID2( $\mathcal{G}, Y, W_{\mathcal{G}}$ )  
2:  $\mathcal{G}' \gets \mathbf{MH}(\mathcal{G}, Y)$   
3:  $Z \gets \{z \in V^{\mathcal{G}'} | \exists y \in Y : (z, y) \in E_d^{\mathcal{G}'}\} \setminus Y$   
4:  $\mathcal{H} \gets$  The induced subgraph of  $\mathcal{G}'$  on its bidirected edges.  
5:  $W_{\mathcal{H}} \gets \{w_e \in W_{\mathcal{G}} | e \in \mathcal{H}\}$   
6:  $V^{\mathcal{H}} \gets V^{\mathcal{H}} \cup \{y^*, z^*\}$   
7: for  $z \in Z$  do  
8:  $E^{\mathcal{H}} \gets E^{\mathcal{H}} \cup \{z^*, z\}$   
9:  $W_{\mathcal{H}} \gets W_{\mathcal{H}} \cup \{w_{\{z^*, z\}} = \sum_y w_{(z, y)}\}$   
10: for  $y \in Y$  do  
11:  $E^{\mathcal{H}} \gets E^{\mathcal{H}} \cup \{y, y^*\}$   
12:  $W_{\mathcal{H}} \gets W_{\mathcal{H}} \cup \{w_{\{y, y^*\}} = \infty\}$   
13:  $E \gets \text{MinCut}(\mathcal{H}, W_{\mathcal{H}}, z^*, y^*)$   
14: return ( $E, \sum_{e \in E} w_e$ )
Noting that the synthetic/simulation results in the main paper were for graphs with a  $\log (n) / n$  sparsity constraint, we begin this section by providing a set of results on the simulated graphs without the sparsity penalty for comparison. Then, we provide information about the causal discovery algorithm used to derive the psychology 'Psych' real-world graph. We also provide experimental results for Problem 2 formulation in Section E.3.
E.1 Additional simulation results without sparsity constraint
The simulation results for graphs generated without the sparsity constraint are shown in Figure 8. These results illustrate monotonic increases in runtime and cost as the number of nodes increases. Our proposed heuristic algorithms (HEID-1 and HEID-2) maintain runtimes less than 0.5 seconds even for 250 nodes. In contrast, the two exact algorithms (MCIP-exact and EDGEID) exceed the three
(b) Solution costs.
(c) Fraction for which runtime of 3 minutes exceeded.
Figure 8: Experimental results (for graphs generated without the sparsity constraint) for runtime, solution costs, fraction of graphs for which no solution was found, and fraction of graphs for which runtime limit of 3 minutes was exceeded. Error bars for runtime and cost graphs indicate 5th and 95th percentiles. Best viewed in color.
minute runtime limit at only 20 nodes, and the MCIP heuristic variants (MCIP-H1 and MCIP-H2) have runtimes which increase exponentially with the number of nodes. These results highlight the efficiency of our proposed heuristic algorithms to find solutions with equivalent cost with significantly faster runtimes.
E.2 Psychology graph discovery
The settings for deriving the putative structure used on the psychology real-world graph are provided in Table 3.
Table 3: Hyperparameter settings for the Structural Agnostic Model used to generate the putative (directed) structure for the 'Psych' real-world dataset.
Parameter	Value
Learning Rate	0.01
DAG Penalty	True
DAG Penalty Weight	0.05
Number of Runs	50
Train Epochs	3000
Test Epochs	800
Mixed Data	True
hlayers	2
dhlayers	2
lambda1	10
lambda2	0.001
dlr	0.001
linear	False
nh	20
dnh	200
(c) Fraction runtime exceeded  $3\mathrm{min}$
Figure 9: Experimental results for runtime, solution costs, fraction of graphs for which no solution was found, and fraction of graphs for which runtime limit of 3 minutes was exceeded. Error bars for runtime and cost graphs indicate 5th and 95th percentiles. Best viewed in color.
E.3 Simulation results for Problem 2 formulation
The experimental setup is exactly as in the main text (the results depicted in Figure 4), except that the probabilities are chosen in the range  $[0.01, 1]$  instead of  $[0.51, 1]$ , and we use the weight mapping corresponding to Problem 2 as described in Lemma 3.2. That is, we map each probability  $p_e$  to the weight  $-\log(1 - p_e)$  in the corresponding edge ID problem.
The simulation results are presented in Figure 9. Runtimes and costs are shown for the subset of graphs for which all algorithms found a solution (to facilitate comparison). Runtimes for each algorithm are shown in Fig. 9a, where it can be seen that our proposed HEID-1 and HEID-2 heuristic algorithms have negligible runtime. In contrast, EDGEID had large runtime variance which depended heavily on the specifics of the graph under evaluation, particularly for graphs with fewer vertices. The costs for each graph are shown in Fig. 9b. Figure 9c shows the fraction of evaluations for which the runtime exceeded 3 minutes (applicable to the exact algorithms). In general, and owing to the sparsity penalty in our graph generation mechanism, the cost of identified solutions falls with the number of vertices. Overall, HEID-1 was both the most consistent in terms of finding a solution, having a short runtime, and achieving a close to optimal cost.