Algorithm 3.

[combinations]=GENERATE_CANDIDATES(K)

Input:
	current nullspace matrix ${\text{K}}_{\text{q}\times {\text{n}}_{\text{ems}}}=\left[\begin{array}{l}{\text{R}}^{(1)}\hfill \\ {\text{R}}^{(2)}\hfill \end{array}\right]$
Output:
	pairs of indices of columns forming candidates (combinations)
1:	irrev + ⇐ (i: ${\text{R}}_{1,\text{i}}^{(2)}>0$ and (∃j: jth reaction is irreversible, ${\text{R}}_{\text{j},\text{i}}^{(1)}\ne 0$)}
2:	irrev − ⇐ (i: ${\text{R}}_{1,\text{i}}^{(2)}<0$ and (∃j: jth reaction is irreversible, ${\text{R}}_{\text{j},\text{i}}^{(1)}\ne 0$)}
3:	rev ⇐ (i: ${\text{R}}_{1,\text{i}}^{(2)}\ne 0$ and (∀j: jth reaction is reversible or ${\text{R}}_{\text{j},\text{i}}^{(1)}=0$)}
4:	{combine columns that can annihilate the element in the current row}
5:	S ⇐ {(ii, jj): (ii, jj) ∈ (irrev+ × irrev−) ∪ ((irrev+ ∪ irrev− ∪ rev) × rev)}
6:	for each (ii, jj) ∈ S do
7:	form candidate column from the pair of columns indexed by (ii, jj)
8:	if candidate satisfies Proposition 4, add (ii, jj) to combinations
9:	end for