I want to know if a specific version (and what´s its name?) of the Set Partitioning Problem is NP-Complete, and if, is NP-Hard?
Problem:
Given a set of elements $e_i$ where $i={1,...,n}$, each with an associated weight $w(e_i)$, we want to find an optimum partition of these elements into a set of $S_j$ subsets where $j={1,...,k}$. Each subset $S_j$ has an objective weight $W_j$. We define a subset's weight as $w(S_j) = \sum_{{e_i}\in{S_j}} w(e_i)$ where $w(S_j)=0$ if and only if $S_j=\emptyset$. Then the objective is to minimize the sum of the absolute deviation of the weights $w(S_j)$ w.r.t. the objective weights $W_j$:
$$min \sum_{j=1}^{k} |w(S_j) - W_j|$$
I do not really need a proof, just an actual answer and if possible, some reference to a paper or an article about it.