# bipartite matching – Predicate variant of Assignment Problem

Given two equally sized sets, $$P$$ of Boolean predicates and $$E$$, I want to decide if there exists a bijective function $$f: P rightarrow E$$, such that
begin{align} forall p in P ; p(f(p)) end{align}

It seems like a special case of having a weight matrix $$W$$ where $$w_{ij}=0$$ if $$P_i(T_j)$$ and otherwise $$1$$, and then solving the general Assignment Problem.
For practical reasons calculating such a weight matrix up front is not desirable, as evaluating $$p$$ is not necessarily cheap.

I have also thought about whether it could be formulated as some variant of Stable Marriage Problem, where the men have a predicate that needs to satisfied instead of a ranked preference list over all women.

I don’t think the proposed problem sounds very exotic, so I might just be looking in a wrong direction.