I have a set $M$, subsets $L_1,...,L_m$ and natural numbers $k,l\leq m$.
The problem is:
Are there $l$ unique indices $1\leq i_1,...,i_l\leq m$, such that
$\hspace{5cm}\left|\bigcap_{j=1}^{l} L_{i_{j}}\right| \leq k$
Now my question is whether this problem is $NP$-complete or not. What irritates me is the two constraints $l$ and $k$ because the NP-complete problems that were conceptually close to it that I took a look on (set cover, vertex cover) only have one constraint respectively that also appears in this problem.
I then tried to write a polynomial time algorithm that looks at which of the sets $L_1,...,L_m$ share more than $k$ elements with other sets but even if all sets would share more than $k$ elements with other this wouldn't mean that their intersection has more than $k$ elements...
This question kind of comes close but in it there is no restriction on the amount of subsets to use and the size of the intersection should be exactly $k$, but maybe this could be useful anyways.
Can somebody further enlighten me ?
