Approximate Minimum Tree Cover in All Symmetric Monotone Norms Simultaneously

We study the problem of partitioning a set of $n$ objects in a metric space into $k$ clusters $V_1,\dots,V_k$. The quality of the clustering is measured by considering the vector of cluster costs and then minimizing some monotone symmetric norm of that vector (in particular, this includes the $\ell_p$-norms). For the costs of the clusters we take the weight of a minimum-weight spanning tree on the objects in~$V_i$, which may serve as a proxy for the cost of traversing all objects in the cluster, but also as a shape-invariant measure of cluster density similar to Single-Linkage Clustering. This setting has been studied by Even, Garg, Könemann, Ravi, Sinha (Oper. Res. Lett.}, 2004) for the setting of minimizing the weight of the largest cluster (i.e., using $\ell_\infty$) as Min-Max Tree Cover, for which they gave a constant-factor approximation. We provide a careful adaptation of their algorithm to compute solutions which are approximately optimal with respect to all monotone symmetric norms simultaneously, and show how to find them in polynomial time. In fact, our algorithm is purely combinatorial and can process metric spaces with 10,000 points in less than a second. As an extension, we also consider the case where instead of a target number of clusters we are provided with a set of depots in the space such that every cluster should contain at least one such depot. For this setting also we are able to give a polynomial time algorithm computing a constant factor approximation with respect to all monotone symmetric norms simultaneously. To show that the algorithmic results are tight up to the precise constant of approximation attainable, we also prove that such clustering problems are already APX-hard when considering only one single $\ell_p$ norm for the objective.