A Configuration-LP Framework for Connected k-Median Clustering

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.