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Approximate Minimum Tree Cover in All Symmetric Monotone Norms Simultaneously

Authors Matthias Kaul , Kelin Luo , Matthias Mnich , Heiko Röglin

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Author Details

Matthias Kaul
  • Hamburg University of Technology, Institute for Algorithms and Complexity, Hamburg, Germany
  • University of Bonn, Germany
Kelin Luo
  • University of Bonn, Germany
  • University at Buffalo, NY, USA
Matthias Mnich
  • Hamburg University of Technology, Institute for Algorithms and Complexity, Hamburg, Germany
Heiko Röglin
  • Universität Bonn, Germany

Funding

  • Kaul, Matthias: Most work done while at the Hamburg University of Technology.
  • Luo, Kelin: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 390685813.
  • Röglin, Heiko: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 459420781 and by the Lamarr Institute for Machine Learning and Artificial Intelligence lamarr-institute.org.

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Matthias Kaul, Kelin Luo, Matthias Mnich, and Heiko Röglin. Approximate Minimum Tree Cover in All Symmetric Monotone Norms Simultaneously. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 57:1-57:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.57

Abstract

We study the problem of partitioning a set of n objects in a metric space into k clusters V₁,...,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 𝓁_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, for example in the context of Multirobot Coverage as studied by Zheng, Koenig, Kempe, Jain (IROS 2005), but also as a shape-invariant measure of cluster density similar to Single-Linkage Clustering.
This problem 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 𝓁_∞) as Min-Max Tree Cover, for which they gave a constant-factor approximation algorithm. 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. One can consider these as the fixed starting points of some agents that will traverse all points of a cluster. 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 𝓁_p norm for the objective.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Clustering
  • spanning trees
  • all-norm approximation

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