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A Configuration-LP Framework for Connected k-Median Clustering

Authors Kushagra Chatterjee , Rojin Rezvan , Ali Vakilian

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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Facility location and clustering
Keywords
  • Connected Clustering
  • k-Median Clustering
  • Configuration LP

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Abstract

We study the connected k-median clustering problem, a clustering problem that augments the classical k-median objective with connectivity constraints. We focus on the overlapping variant of the problem, where clusters are allowed to share vertices. In addition to a metric space (V,d), the input contains a connected graph G on the same vertex set V of size n. The goal is to select at most k centers C and assign vertices to them so as to minimize the k-median cost (i.e., ∑_{v ∈ V} d(v,C)), subject to the constraint that each cluster induces a connected subgraph of G. Since the metric space and the connectivity graph are independent, the problem is significantly more challenging than standard clustering. Eube et al. [Eube et al., 2025] showed that even the assignment version is Ω(log n)-hard to approximate and gave approximation algorithms with guarantees depending polynomially on k.
We develop a configuration-LP-based framework that combines covering LP techniques with a rooted minimum-density oracle. For the assignment version, we obtain an O(log² n)-approximation. For the general version, we develop a bicriteria framework that opens O(klog n) centers while achieving an O(log² n)-approximation in cost.

Cite As Get BibTex

Kushagra Chatterjee, Rojin Rezvan, and Ali Vakilian. A Configuration-LP Framework for Connected k-Median Clustering. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.22

Author Details

Kushagra Chatterjee
  • Indian Statistical Institute, Kolkata, India
Rojin Rezvan
  • Virginia Tech, Blacksburg, VA, USA
Ali Vakilian
  • Virginia Tech, Blacksburg, VA, USA

Funding

  • Vakilian, Ali: Supported in part by NSF CAREER Award CCF-2542925.

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