Consider the vertices of an $n$-dimensional cube. The distance between two vertices is measured as the minimum number of edges between the two vertices. Now consider a subset of these vertices. If we call the total set of vertices as $T$ and the subset as $S$ then our purpose is to partition $S$ into two sets $A$ and $B$ and for both of these sets find vertices $x_A$ and $x_B$ from $T$ such that the sum total of the distance between $x_A$ and the vertices of $A$ and $x_B$ and the vertices of $B$ should be a minimum.

How to approach this question?

Given that we know the distance relation for every pair of vertex, is it possible to know the minimum distance through some simple calculation?