Smoothed analysis of the Partition problem

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?

Smoothed analysis, as I have seen it, has typically been used for analyzing optimization problems, not decision problems. The partition problem is a decision problem, which introduces the challenge you mentioned. Perhaps one could consider the optimization variant of the partition problem: given $$n$$ numbers with a sum of $$2S$$, find the subset whose sum is as close to $$S$$ as possible. I don't know whether the smoothed complexity of that optimization problem would be interesting or illuminating, but it eliminates the problems you articulate in the question.