Will this reduction of Exact Cover into Subset-Sum fail due to a potential false positive?
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29-09-2020 - |
Question
After removing multi-sets and sets that have elements that don't exist in $S$.
$S$ = $[9,6,7,4,5,1,8]$
$C$ =$[[9,6,7],[4,5],[1,8]]$
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 $[1],[1]s...$ 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 $C$ = $[[14],[41],[85]]$
- Run Algorithm and get the solution
Question
Will this reduction of Exact Cover into Subset-Sum every yield a false positive?
Solution
If you're not really sure whether a reduction will work, it probably won't. Whenever you are making a reduction you should always have a plan of how to prove it correct.
In this case, we're looking to see whether $1^2+2^2+3^2+...+n^2$ can be written as a sum of squares in some other way (which would be a false positive).
We have that $4^2=2^2+2^2+2^2+2^2$. This is enough to construct a counterexample.
Let $S=\{1,2,3,4,5,6,7\}$ and $C=\{\{1,2\},\{2,3\},\{2,5\},\{2,6\}\,\{2,7\},\{1,3,4,5,6,7\}\}$.
There is no exact cover (the only way to get $4$ is to take $\{1,3,4,5,6,7\}$ but then we cannot take any other set since all of them overlap).
However, we have that:
$$1^2+2^2+3^2+4^2+5^2+6^2+7^2=(1^2+2^2)+(2^2+3^2)+(2^2+5^2)+(2^2+6^2)+(2^2+7^2)$$
Thus, we can conclude the reduction does not work.