Given:
$G:=\{V= \{v_1,\ ...\ v_n\},E\subset V\times V\}, n<\infty$, a symmetric complete, simple graph
$w:=\ \ E \ni e_{ij}\mapsto \mathbb{R}^+$, a weight function for the edges of $G$
$K:=\{c_1,\ ...\ c_m\},\ 2\le m\ll n$, a set of colors to be assigned to the vertices of $G$
Question:
Are there any algorithms known, that color the vertices in $V$ with colors from $K$ in a way that minimizes some norm of the vector of nearest color-neighbor distances;
i.e. if $c(j)$ is the color assigned to $v_j$ and, $\mu_{ik} := \min\limits_{c(v_j)=c_k}w_{ij}$ the least weight of the edges that connect vertex $v_i$ to a vertex $v_j$ of color $c_k$, are algorithms known, that minimize $\|(\mu_{1,1}\ ,\ ...,\ \mu_{1,k}\ ,\ ...,\ \mu_{n,1},\ ...,\ \mu_{n,k})\|$?
In the context of this question a vertex is considered to be none if its neighbors, but feel free to assume the contrary.
