Algorithm 1: IV-GIN

Input: Treatment X, outcome Y, and set of observed variables $\mathbf{O}$ .  Output: Set of candidate $\mathbf{C}$ and its corresponding conditional set $\mathbf{Conset}$ .   1: Initialize the set of candidate IVs: $\mathbf{C}=\varnothing$ , the conditional set: $\mathbf{Conset}=\varnothing$ , the length of conditional set: $\mathrm{ConsetLen}=0$ , and $\mathbf{Tag}=\mathbf{O}$ ;   2: while $\mathrm{ConsetLen}<\left|\mathbf{Tag}\right|$ do   3:    for each variable $Z_i\in \mathbf{C}$  do   4:     repeat   5:         Select a subset $\mathbf{W}$ from $\mathbf{O}\setminus Z_i$ such that $\mathbf{W}=\mathrm{ConsetLen}$ ;   6:         if  $\left[Z\right|\left|\mathbf{W}\right]$ follows the IV-GIN condition then   7:           Add $Z_i$ into $\mathbf{C}$ , and delete $Z_i$ from $\mathbf{Tag}$ ;   8:           Set $\mathbf{Conset}\left(Z_i\right)=\mathbf{W}$ ;   9:           Break the repeat loop of line 4;   10:         end if   11:      until all subsets with length $\mathrm{ConsetLen}$ in $\mathbf{O}\backslash Z_i$ are selected;   12:    end for   13:     $\mathrm{ConsetLen}=\mathrm{ConsetLen}+1$ ;   14: end while   15: Return: $\mathbf{C}$ and $\mathbf{Conset}$