Algorithm 1

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

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

For $k,l=1,\cdots ,m$ do begin

If $F_{k\mathrm{,}l}\left(H\mathrm{,}V\right)$ is satisfiable,

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

end

output “failure”.