Approximate Minimum Tree Cover in All Symmetric Monotone Norms Simultaneously

2025-01-09

AI Summary

This paper studies the clustering problem of partitioning $n$ objects in a metric space into $k$ clusters, aiming to approximately minimize the total cost—defined as the symmetric monotone norm (including all $ell_p$ norms) applied to the vector of minimum spanning tree weights of the clusters—subject to the constraint that each cluster must contain at least one designated hub. We propose the first unified combinatorial algorithm that achieves a constant-factor approximation for *all* such norms in polynomial time, without relying on numerical optimization or learning—a breakthrough over prior norm-specific algorithms. We further prove that the problem is APX-hard under multiple-hub constraints, establishing a tight theoretical bound on approximability. Our algorithm scales efficiently to instances with tens of thousands of points, solving them in under one second, thus achieving both theoretical optimality and practical scalability.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Clustering
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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, K""onemann, 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.

Problem

• Clustering
• Symmetric Tree Cover
• APX-Hard Problem

Innovation

• Approximate Minimum Tree Covers
• Cluster Optimization Algorithm
• APX-hard Problem Evaluation