After removing multi-sets and sets that have elements that don't exist in $S$.
$S$ = $[9,6,7,4,5,1,8]$
Transform the values in $C$ of the shared index values with $S$. (This must be done before touching $S$)
$C$ = $[[1,2,3],[4,5],[6,7]]$
And do the same for $S$
$S$ = $[1,2,3,4,5,6,7]$
Square each $x$ integer in both $S$ and $C$
$f(x)$ = $x^2$, $x ∈ S$ then $C$
$S$ = $[1, 4, 9, 16, 25, 36, 49]$
$C$ = $[[1,4,9],[16,25],[36,49]]$
Then remove all sets with repeating sums to prevent false positives. This means no $,s...$ that could be used, to sum up to the total sum of $S$, This also means $[1,4,9]$ or $[4,9,1]$. (leave one though!)
- After the transformation is complete use subset-sum solver and define total-sum as $140$ (total-sum of $s$)
- Define list of integers as a summed up set of $C$ = $[,,]$
- Run Algorithm and get the solution
Will this reduction of Exact Cover into Subset-Sum every yield a false positive?