What is known about mixed integer linear programs of the form $$ \begin{array}{rll} \max & c^T \mathbf{x} + d^T \mathbf{y} & \\ \text{subject to:} & a_i^T \mathbf{x} + b_i^T \mathbf{y} \leq p_i, & i=1,2,\dotsc,n \\ & \mathbf{y} \in \{ 0, 1 \}^{q} \end{array} $$
where for almost every fixed $\mathbf{x}$, there exists at most one $\mathbf{y}$ such that $(\mathbf{x}, \mathbf{y})$ is feasible?
I am particularly interested in the methods of solving those programs using this additional information. Could there be improvements in the branch and bound algorithm?
Any partial, similar, or more general results are also welcome - for example, anything about the feasibility region. If there are results that just assume some bound on the number of different feasible $\mathbf{y}$ after fixing $\mathbf{x}$, it would too be useful.