Algorithm 2

Input: $H\in {\mathbb{N}}^m\mathrm{,}V\in {\mathbb{N}}^n$

W.l.o.g assume: $\forall i:h_i\in \left[1,n\right]$ , $\forall j:v_j\in \left[1,m\right]$ , ${\displaystyle {\sum }_i{h}_i}={\displaystyle {\sum }_j{v}_j}$ .

For $\min \left(W\right),\min \left(E\right)=1,\cdots ,m$ and

$\min \left(N\right),\min \left(S\right)=1,\cdots ,n$ do begin

If ${\gamma }_{2L\mathrm{,}Geo1}^{\mathrm{1,1}}\left(H\mathrm{,}V\right)$ or ${\gamma }_{2L\mathrm{,}Geo2}^{\mathrm{1,1}}\left(H\mathrm{,}V\right)$ or ${\gamma }_{2L\mathrm{,}Geo3}^{\mathrm{1,1}}\left(H\mathrm{,}V\right)$ or ${\gamma }_{2L\mathrm{,}Geo4}^{\mathrm{1,1}}\left(H\mathrm{,}V\right)$ is satisfiable,

then output $P=\overline{A\cup B\cup C\cup D}$ and halt.

end

output “failure”.