I am studying smoothed analysis and trying to apply it to the Partition decision problem: given $n$ real numbers with a sum of $2 S$, decide whether there exists a subset with a sum of exactly $S$.
The common definition of the smoothed runtime complexity of an algorithm is: given $n$ and $\sigma$, the smoothed runtime of an algorithm is the maximum, over all inputs of size $n$, of the runtime on the input when it is perturbed by a perturbation of size $\sigma$, e.g. by adding to each input a number selected randomly from a normal distribution with standard deviation $\sigma$, or from any distribution with support $[0,\sigma]$.
If I apply this definition to the Partition problem, it seems that for any $\sigma > 0$, the runtime complexity is $O(1)$, since for any random noise that is added to the original numbers - no matter how small - the answer is "no" with probability 1.
This is strange, since in the more common examples of smoothed analysis, the runtime complexity depends on $\sigma$, and here it does not.
Is there something I misunderstood? What is the smoothed runtime complexity of Partition?