# Is is possible to create a SUBSET-SUM instance that each subset is “unique”?

Given a SUBSET-SUM instance $$S$$ with a weight $$W$$, is it possible to create, in polynomial time, a new non-empty instance $$T$$ (at most the same length as $$S$$) with weight $$M$$, that for each non-empty subset of set $$T$$ and its sum $$s$$, it has a unique $$(s-M)^2$$, and that a subset of $$T$$ adds up to $$M$$ iff a subset of $$S$$ adds up to $$W$$?

• Is $M$ an input or an output? Can it be freely chosen by the algorithm? – D.W. Sep 16 at 4:34
• If there exists such a reduction, then the variant of subset sum problem where each subset has a unique sum would be NP-complete. Unfortunately, no one has proved or disproved it. – xskxzr Sep 16 at 10:41
• @D.W. It's an output, it only depends on $S$ and $W$. – DUO Sep 16 at 14:25
• @xskxzr I edited it, so it may be easier to prove (or disprove). – DUO Sep 16 at 15:03