What's complexity of this set problem which looks like "Linear Programming"? I came up with a problem below, which looks like a linear programming problem:


  Given $n$ sets $S_{1}, S_{2},..., S_{n}$, with constraints of :
  $$
\forall i=1, 2, 3,...,n\space\space \left | S_{i} \right |=k;\\
\left | S_{j_{11}}\bigcup S_{j_{12}} \right |\leqslant k_{1};\\
\left | S_{j_{21}}\bigcup S_{j_{22}} \right |\leqslant k_{2};\\
...\\
\left | S_{j_{m1}}\bigcup S_{j_{m2}} \right |\leqslant k_{m}.\\
$$
  $j_{11},j_{21},...,j_{m1},j_{12},j_{22},...,j_{m2}$ are integers between $1$ and $n$. $k,k_{1},k_{2},...,k_{m}$ are constant positive integers. 
  The questions is:
  For a positive integer $q$,Do such $S_{1}, S_{2},..., S_{n}$ exsit, such that $\left| \bigcup_{i}^{n}S_{i} \right |=q$?


I came up with this problem in order to judge complexity of another graph problem. But I failed to find any relative research. So has complexity of this kind of problem been studied? Is this problem NP-Complete or just P? How to judge this problem? Thanks.
By the way, what if $k=2,k_{1}=k_{2}=...=k_{m}=3$?
If I'm not mistaken, the case you specify (with $k =2$, $k_i = 3$) is in P. 

Note that the condition $|S \cup S'| \le 3$ is equivalent to saying that $S, S'$ intersect (since $k=2$). Thus, if you think of the $S_i$ as vertices, then the constraints specify a graph $G$, and your problem reduces to finding a graph $H$ such that $G$ is the line graph of $H$. 

Verifying that $G$ is indeed a line graph and then finding $H$ can be done in linear time.