I am looking for a reference for a result that I am aware of. Let me describe the result.
Given a polygon $C$ and polygons $p_1,\ldots,p_n$, it can be decided in NP time, if $p_1,\ldots,p_n$ can be placed under translation into $C$.
The NP-algorithm goes as follows. guess for each pair of one line segment and one vertex their combinatorics. In other words, Is the point above or below the line spanned by the segment. Build a Linear program (LP) to check if all of those constraints can be met. Here the LP has $2n$ variables (two for each translation vector for each $p_i$) and each constraint is linear with up to four variables.
Note that one cannot just guess the coordinates, as we do not know a priori, they have polynomially bounded number of precision required.
I know the result from my PHD advisor Günter Rote, but I could not find a reference for this result.