Kth largest subset for small K
There is a simple algorithm whose complexity is $O(Kn)$. The algorithm computes, for each $m \leq n$, the $K$ largest sums of the first $m$ numbers. Given the list for $m$, the list for $m+1$ is computed by adding the $(m+1)$st element to the old list, merging the two lists, and keeping only the top half. Since merging sorted lists can be done in linear time, the resulting complexity is $O(Kn)$.