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.

翻译：在最小半径和聚类问题中，给定度量空间中有限点集X（含n个点），目标是找出至多k个以X子集为中心的聚类，使得X中每个点都被分配到某个聚类中，并最小化各聚类半径之和。Gibson等人（SWAT 2008）证明，即使在加权平面图诱导的度量空间和常倍维数度量空间上，该问题也是NP困难的。本文研究无向图诱导度量空间上最小半径和问题的参数化复杂度，并区分加权图度量（正边权）与非加权图度量（单位边权）。对于加权图度量：我们证明当参数化为组合参数k（聚类数量）与Δ（聚类成本）时，最小半径和问题在加权二分图诱导的度量空间上是W[1]-难的。进一步研究问题的结构参数化复杂度：Drexler等人（arXiv:2310.02130）表明，当参数化为树宽时，最小半径和问题在加权图诱导度量空间上存在XP算法，并询问能否改进为固定参数可解性。我们首先否定回答该问题，并进一步证明即使参数化为顶点覆盖数加k，最小半径和问题在无向加权二分图诱导度量空间上仍保持W[1]-难性。随后关注稠密图的参数，证明在团和完全二分图上，当参数化为k+Δ时最小半径和问题仍是W[1]-难的。在正面结果方面，我们利用已知的以树宽为参数的XP算法，证明当参数化为树宽加Δ时最小半径和问题是FPT的。