Here's the problem.
Suppose there are $n$ users, each with an value $\sigma_i$. There are $m$ tasks, and we want to assign each user a task. Suppose for each task $j$, the set of users assigned to it is denoted by $S_j$.
Each task has a reward $v_j$. Accomplishing each task $j$ needs to satisfy that $\frac{\sum_{i \in S_j}\sigma_i}{|S_j|^2} \leq \theta$, where $\theta$ is a fixed constant. Let $u_j \in \{0,1\}$ denote if the task $j$ is accomplished ("1" means accomplished, and "0" means not).
The objective of the problem is to find a user assignment method so as to maximize the total obtained value of tasks, that is: $$\underset{\{S_j\}}{\textrm{max}} ~~ \sum_{j=1}^m v_j* u_j$$
I'm not sure which NPC problem can be reduced to this problem. Can someone help me?
A formal formalization of the problem is: \begin{align} \textrm{max} ~~ & v_j * y_j \notag \\ \textrm{s.t.}~~ & \sum_{i=1}^n x_{i,j} \sigma_i^2 \leq \theta (\sum_{i=1}^n x_{i,j})^2, \forall j \notag \\ & \sum_{j=1}^m x_{i,j}=1, \forall j \notag \\ & y_j \in \{0,1\} \notag \\ & x_{i,j} \in \{0,1\} \notag \\ \end{align}
We can see that the problem is nonlinear integer programming, which is NP-hard.