Approximate Minimum Tree Cover in All Symmetric Monotone Norms Simultaneously

TL;DR

This work studies All-Norm Tree Cover, where a metric-space partition into $k$ clusters minimizes the all-$\ell_p$-norm of the MST-based cluster costs. It delivers a purely combinatorial polynomial-time algorithm that achieves a constant-factor approximation simultaneously for all monotone symmetric norms, and extends this to the All-Norm Tree Cover with Depots via a layered, depot-aware scheme. The authors introduce strong-optimality concepts, provide a refinement to guarantee exactly $k$ trees, and prove hardness results (APX-hardness for single norms and NP-hardness of approximation for the depot variant), situating the results within the landscape of predictable, norm-agnostic clustering. They also demonstrate practical efficiency on large instances (up to about $10^4$ points), highlighting the approach’s potential for flexible, tree-based clustering in real-world settings.

Abstract

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.

Figures (9)

• Figure 1: Two instances of the All-Norm Tree Cover problem. \ref{['fig:CounterexamplesOneNormGood']} is an instance where an optimal $\ell_1$-norm solution does not return a good approximation in the $\ell_\infty$-norm; \ref{['fig:CounterexamplesInftyNormGood']} is an instance where an optimal $\ell_\infty$-norm solution does not return a good approximation in the $\ell_1$-norm.
• Figure 2: Example instance where the algorithm of Even et al. even2004min does not return a good approximation in the $\ell_1$-objective. The left figure is an example of the graph metric when $k=4$ and the right figure is the final solution obtained by the rooted-tree-cover algorithm by Even et al. even2004min.
• Figure 3: Example instance where the algorithm of Even et al. does not return a good approximation in the $\ell_1$-objective. The instance consists of $n$ identical copies $H_i$ of the $4$-star where all edges have length $R$ and the copies are pairwise at distance $2R$. For $k=5n-1$, the instance has $OPT_\infty = OPT_1 = R$, but the algorithm will return $2n$ trees, each of weight $2R$. Observe that the partition requirement in \ref{['alg:item:Decomposition']} would also be fulfilled if the trees are not further partitioned and kept at size $4R$, however the algorithm of Even et al. produces the former solution. Both solutions do not achieve a good approximation in the $\ell_1$ objective.
• Figure 4: Illustration of how the algorithm's solution and some optimal solution can align against each other. The algorithm's partition of the graph into connected component is indicated by solid zones, the further subdivision of the large components by dotted zones. The trees of the optimal solution are drawn, and all edges crossing the boundary of a connected component are dashed.
• Figure 5: Visualisations of the results of our implementation of the non-depot tree cover algorithm. Inaccessible sections of the grid are marked in black; other colors represent the computed clusters.

Theorems & Definitions (42)

• Definition 1: All-norm clustering problem
• Theorem 1
• Theorem 2
• Theorem 3
• Definition 2: Tree cover
• Definition 3: Tree cover with depots
• Definition 4: $\ell_p$- Tree Cover (resp. $\ell_p$- Tree Cover with Depots )
• Definition 5: All-Norm Tree Cover Problem (resp. All-Norm Tree Cover Problem with Depots)
• Lemma 1: Kou et al. kouFastAlgorithmSteiner1981
• proof