# Hardness result or reference for a set partition problem

I'm wondering if the following problem is (or has been proven to be) NP-Complete.

Input: integer $$n\ge0$$, set $$S_1,S_2,\ldots,S_{2n}$$, set $$T_1,T_2,\ldots,T_n$$.

Accept iff: there exists $$\{a_i,b_i\},i=1,2,\ldots,n$$ such that $$\{a_1,\ldots,a_n,b_1,\ldots,b_n\}=\{1,2,\ldots,2n\}$$ and $$S_{a_i}\cup S_{b_i}=T_i,S_{a_i}\cap S_{b_i}=\emptyset$$.

That is, we want to partition $$\{S_i\}$$ as $$n$$ group of two sets, and the $$i$$-th group exactly covers $$T_i$$.

Towards algorithm, we can construct a colored graph with vertex set $$\{v_1,\ldots,v_{2n}\}$$ and $$v_i,v_j$$ is connected with color $$k$$ iff $$S_i\cup S_j=T_k,S_i\cap S_j=\emptyset$$. Then the problem asks for a perfect matching of distinct color.

I suspect the problem was considered before; or there is some other problem closely related. But all I can find is the comment in this problem.