Algorithm 2 Primal-dual heuristic for finding an HSF
1:	Initialization
2:	$F_k\leftarrow \varnothing$, ${\mathcal{C}}_k\leftarrow \{\left\{v\right\}:v\in V_k\}$ for $k=1,2$
3:	All vertices are unmarked.
4:	All dual variables are set to zero.
5:	$activ{e}_k\left(\left\{v\right\}\right)\leftarrow 1$, $\forall v\in V_k$, for $k=1,2$
6:	$activ{e}_k\left(\left\{d_k\right\}\right)\leftarrow 0$, for $k=1,2$
7:	Main loop
8:	while there exists any active component in ${\mathcal{C}}_1$ do
9:	for $k=1,2$ do
10:	Find an edge $e_k=(i,j)\in E_k$ with $i\in C_i,j\in C_j$ where $C_i,C_j\in {\mathcal{C}}_k,C_i\ne C_j$ that minimizes ${\epsilon }_k=\frac{\{W_k\times cos{t}_{ij}^k-dua{l}_k\left(i\right)-dua{l}_k\left(j\right)\}}{activ{e}_k\left(C_i\right)+activ{e}_k\left(C_j\right)}$.
11:	end for
12:	Let $\Re :=\{R:activ{e}_k\left(R\right)=1,\mathrm{One}\mathrm{of}\mathrm{subsets}\mathrm{of}R\mathrm{contains}{d}_2,\forall R\in {\mathcal{C}}_1\}$.
13:	Find $\overline{R}\in \Re$ that minimizes ${\epsilon }_3=bound\left(\overline{R}\right)-Y_1\left(\overline{R}\right)$
14:	${\epsilon }_{min}=min({\epsilon }_1,{\epsilon }_2,{\epsilon }_3)$
15:	for $k=1,2$ do
16:	for $C\in {\mathcal{C}}_k$ do
17:	$Y_k\left(C\right)\leftarrow Y_k\left(C\right)+{\epsilon }_{min}\times activ{e}_k\left(C\right)$
18:	$dua{l}_k\left(v\right)\leftarrow dua{l}_k\left(v\right)+{\epsilon }_{min}\times activ{e}_k\left(C\right),\forall v\in C$
19:	if $k=1$ then
20:	$bound\left(C\right)\leftarrow bound\left(C\right)+{\epsilon }_{min}\left|Children\left(C\right)\right|$
21:	end if
22:	end for
23:	end for
24:	if ${\epsilon }_{min}={\epsilon }_k$ for $k=1$ or 2 then
25:	$F_k\leftarrow \left\{e_k\right\}\cup F_k$
26:	${\mathcal{C}}_k\leftarrow {\mathcal{C}}_k\cup \{C_i\cup C_j\}-C_i-C_j$
27:	$Y_k\left(\{C_i\cup C_j\}\right)=Y_k\left(C_i\right)+Y_k\left(C_j\right)$
28:	if $k=1$ then
29:	$Bound(C_i\cup C_j)\leftarrow bound\left(C_i\right)+bound\left(C_j\right)$
30:	end if
31:	if $d_k\in \{C_i\cup C_j\}$ then
32:	$activ{e}_k(C_i\cup C_j)\leftarrow 0$
33:	if $k=1$ then
34:	$activ{e}_2\left(C\right)\leftarrow 0\forall C\in Children(C_i\cup C_j)$
35:	end if
36:	else
37:	$activ{e}_k(C_i\cup C_j)\leftarrow 1$
38:	end if
39:	else
40:	$activ{e}_1\left(\overline{C}\right)\leftarrow 0$
41:	Mark all of the vertices of $\overline{C}$ with the label $\overline{C}$.
42:	end if
43:	end while
44:	Pruning
45:	$P_k\leftarrow \mathrm{all}\mathrm{vertices}\mathrm{that}\mathrm{are}\mathrm{only}\mathrm{connected}\mathrm{to}{d}_k\mathrm{for}k=1,2$
46:	$P_c\leftarrow \mathrm{the}\mathrm{vertices}\mathrm{that}\mathrm{are}\mathrm{connected}\mathrm{to}\mathrm{both}{d}_1\mathrm{and}{d}_2$
47:	Let $T_k$ be the minimum spanning tree of $P_k\mathrm{for}k=1,2$.
48:	Let $C\left(T_k\right)$ be the sum of edge costs present in $T_k$
49:	while$P_c$ is not empty do
50:	Find the shortest edge $e_k$ that connects a vertex in $P_c$ and a vertex in $P_k$ for each k.
51:	if $C\left(T_1\right)\le C\left(T_2\right)$ then
52:	Add $e_1$ to $T_1$, remove the corresponding vertex from $P_c$, and add it to $P_1$.
53:	else
54:	Add $e_2$ to $T_2$, remove the corresponding vertex from $P_c$ and add it to $P_2$.
55:	end if
56:	end while