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 $k=2,k_{1}=k_{2}=...=k_{m}=3$?
What's complexity of this set problem which looks like Linear Programming?
set theory;cc.complexity theory;np complete