Input: A set of $n$ Users $U=\{u_1, ..., u_n\}$ and a set of $m$ products $I=\{i_1, ..., i_m\}$. Associated with each pair $u \in U$ and $i \in I$  is the probability $p_{u,i}$ of $u$ purchasing the product $i$.Task: Assign each user exactly $k$ products so that the following objective function is maximized:  $$\sum_{i \in I(U)} ({1-\prod_{u\in U(i)}{(1-p_{u,i})}})$$  Where $I(U)\subseteq I$ is the set of products assigned to some user, and $U(i) \subseteq U$ is the set of users to whom a product $i$ is assigned.Question: Is this problem NP-Hard?
Hardness of a special case of maximum matching
complexity theory;reductions;np hard;matching