# Combinatorial Problem similar in nature to a special version of max weighted matching problem

I have a problem and want to know if there is any combinatorial optimization that is similar in nature to this problem or how to solve this special version of the max weight matching problem.

I have a general graph $$G(\mathcal{V},\mathcal{E},\mathcal{W})$$. I want to find a maximum weight matching of the graph $$G$$ that must cover a certain subset of vertices and has a specific size. For example, if I have a graph with eight vertices, I want to find a max weighted matching that must cover the subset of vertices $$\mathcal{V}’=\{1,2,3\}$$ and the size of the matching is $$\lceil{|\mathcal{V}’|/2}\rceil$$. So one more vertex needs to be chosen that maximizes the weighted matching. How to find the optimal solution in polynomial time if possible?