I have a weighted undirected graph G=(V,E,W), I want to find an independent set S of V, such that if we sort the vertices of S in increasing order of their weights, the cumulative sum of the weights is maximized. In other words, if we suppose without loose of generality that $S={v_1,...,v_h}$ are the vertices of $S$ sorted in increasing order of their weights, then we want to maximize $\sum_{i=1}^{h} \sum_{j=1}^{i}w_j$, where $w_j$ is the weight of vertex $v_j$.
This problem is NP-hard since it can be reduced to the maximum independent set problem when all the weights are equal, but I can't find in the literature a paper dealing with this problem, can you please share any work on this problem or related problems.
For instance if the weights of my independent set are {3,6} than the cumulative sum is 3+(3+6)=12, and if the weights are {1,1,2,3} than my cumulative sum is: 1+(1+1)+(1+1+2)+(1+1+2+3)=14. So an independent set of maximum weight is not necessary yielding the maximum cumulative sum of weights.
