Average distance in colored graphs

For a graph G where the vertices are coloured, the coloured distance of G is defined as the sum of the distances between all unordered pairs of vertices having different colours. Then for a fixed supply s of colours, ds(G) is defined as the minimum coloured distance over all colourings with s. This generalizes the concepts of median and average distance. In this paper we explore bounds on this parameter especially a natural lower bound and the particular case of balanced 2-colourings (equal numbers of red and blue). We show that the general problem is NP-hard but there is a polynomial-time algorithm for trees.