On the Parameterized Complexity of Min-Sum-Radii

2026-05-21

AI Summary

This study investigates the parameterized complexity of the Min-Sum-Radii clustering problem under graph-induced metrics. Although the problem is known to be NP-hard, its tractability with respect to various structural graph parameters remained unclear. We establish that the problem remains W[1]-hard even when parameterized jointly by the vertex cover number and the number of clusters, and we further demonstrate its computational intractability on cliques and complete bipartite graphs. On the positive side, we design a fixed-parameter tractable (FPT) algorithm showing that the problem becomes solvable in FPT time when parameterized by the sum of treewidth and the target cost. This work systematically delineates the complexity landscape of Min-Sum-Radii across diverse graph metrics and parameter combinations.

Abstract

In the Min-Sum-Radii (MSR) clustering problem, we are given a finite set X of n points in a metric space. The objective is to find at most k clusters centered at a subset of these points such that every point of X is assigned to one of the clusters, minimizing the sum of the radii of the clusters. The problem is known to be NP-hard even on metrics induced by weighted planar graphs and metrics with constant doubling dimension, as shown by Gibson et al. (SWAT 2008). In this work, we investigate the parameterized complexity of MSR on metrics induced by undirected graphs. We distinguish between weighted graph metrics (with positive edge weights) and unweighted graph metrics (where all edges have unit weight). Weighted Graph Metrics: We show that MSR is W[1]-hard on metrics induced by weighted bipartite graphs, when parameterized by the combined parameter k (the number of clusters) and Delta (the cost of the clustering). We then investigate the structural parameterized complexity of the problem. Drexler et al. (arXiv:2310.02130) showed that the MSR problem admits an XP algorithm on metrics induced by weighted graphs when parameterized by treewidth, and asked whether this can be improved to fixed-parameter tractability. We first answer their question in the negative, and more strongly show that MSR stays W[1]-hard on metrics induced by undirected weighted bipartite graphs when parameterized by the vertex cover number plus k. We then turn our attention to parameters for dense graphs and show that MSR remains W[1]-hard when parameterized by k+Delta even on cliques and complete bipartite graphs. On the positive side, we employ the known XP algorithm parameterized by treewidth, to show that the MSR problem is FPT when parameterized by the parameter treewidth plus Delta.

Problem

• Min-Sum-Radii
• parameterized complexity
• clustering
• graph metrics
• W[1]-hardness

Innovation

• parameterized complexity
• Min-Sum-Radii clustering
• W[1]-hardness
• treewidth
• fixed-parameter tractability