Partially defined boolean function

Consider boolean function $ f(x_{1}, x_{2}, \dots, x_{n})$ . The value of $ f$ is defined on some set of inputs, and some inputs are undefined (let us label undefined value with $ ?$ ). It is possible to set $ f$ ‘s value on some inputs, no matter if they were initially defined.

The problem is to set some of the inputs such that:

  • all initially undefined inputs become defined (i. e. no $ ?$ in function truth table);
  • $ f$ is monotone in the common sense.

The optimization part is to use the minimum possible assignments. How can one find an optimal way in the time $ \mathcal{O} \left( p(n) \left( 2^{n} \right)^{2} \right)$ , where $ p(n)$ is polynomial?