We first consider the search version of the subset sum problem: Given a set $S$ of $n$ naturals, find a subset of $S$ that sums to exactly $W$. My question concerns this problem, with an additional restriction on set $S$: For all possible values of $X$, there exists at most one subset that sums to $X$ (in other words, no two subsets of $S$ sum to the same value). Can we find a polynomial time algorithm for this problem given this restriction?

My thoughts: The reason I think this may be possible is because this is an extremely strong restriction on $S$. Most sets are very far from having this property. The construction of an algorithm would likely begin with a strong characterization of the sets that even have this property, and work from there. However, I'm having trouble proving strong theorems about the sets that obey this restriction.

Edit: This question has been reasked on TCS.

We first consider the search version of the subset sum problem: Given a set $S$ of $n$ naturals, find a subset of $S$ that sums to exactly $W$. My question concerns this problem, with an additional restriction on set $S$: For all possible values of $X$, there exists at most one subset that sums to $X$ (in other words, no two subsets of $S$ sum to the same value). Can we find a polynomial time algorithm for this problem given this restriction?

My thoughts: The reason I think this may be possible is because this is an extremely strong restriction on $S$. Most sets are very far from having this property. The construction of an algorithm would likely begin with a strong characterization of the sets that even have this property, and work from there. However, I'm having trouble proving strong theorems about the sets that obey this restriction.

We first consider the search version of the subset sum problem: Given a set $S$ of $n$ naturals, find a subset of $S$ that sums to exactly $W$. My question concerns this problem, with an additional restriction on set $S$: For all possible values of $W$, there exists at most one solution to this problem (in other words, no two subsets of $S$ sum to the same value). Can we find a polynomial time algorithm for this problem given this restriction?

My thoughts: The reason I think this may be possible is because this is an extremely strong restriction on $S$. Most sets are very far from having this property. The construction of an algorithm would likely begin with a strong characterization of the sets that even have this property, and work from there. However, I'm having trouble proving strong theorems about the sets that obey this restriction.

We first consider the search version of the subset sum problem: Given a set $S$ of $n$ naturals, find a subset of $S$ that sums to exactly $W$. My question concerns this problem, with an additional restriction on set $S$: For all possible values of $X$, there exists at most one subset that sums to $X$ (in other words, no two subsets of $S$ sum to the same value). Can we find a polynomial time algorithm for this problem given this restriction?

My thoughts: The reason I think this may be possible is because this is an extremely strong restriction on $S$. Most sets are very far from having this property. The construction of an algorithm would likely begin with a strong characterization of the sets that even have this property, and work from there. However, I'm having trouble proving strong theorems about the sets that obey this restriction.