AI 中文总结
针对允许顶点共享的连通k-中位数聚类问题，提出结合配置-LP、覆盖LP和有根密度预言机的框架，分别得到分配版本O(log²n)近似比和一般版本O(k log n)中心数下的O(log²n)代价近似比。
AI 中文摘要
我们研究连通k-中位数聚类问题，该问题在经典k-中位数目标的基础上增加了连通性约束，我们关注该问题的重叠变体，其中聚类允许共享顶点。除了度量空间(V,d)外，输入还包含一个顶点集同为V、规模为n的连通图G。目标是选择至多k个中心C并将顶点分配给它们，以最小化k-中位数代价（即∑_{v∈V} d(v,C)），同时满足每个聚类诱导出G的连通子图的约束。由于度量空间与连通图相互独立，该问题比标准聚类问题更具挑战性。Eube等人[1]证明，即使是分配版本的问题，其近似比也具有Ω(log n)的难度，并给出了保证依赖于k的多项式的近似算法。我们开发了一种基于配置-LP的框架，将覆盖LP技术与有根最小密度预言机相结合。对于分配版本，我们得到了O(log²n)的近似比；对于一般版本，我们开发了一种双准则框架，该框架打开O(k log n)个中心，同时实现O(log²n)的代价近似比。我们的结果为处理聚类问题中的连通性约束提供了一种不同的基于LP的方法，并证明配置LP、覆盖LP和有根密度预言机可以有效结合，以在图论约束下为聚类目标获得近似保证。
英文摘要
We study the \emph{connected $k$-median} clustering problem, a clustering problem that augments the classical $k$-median objective with connectivity constraints. We focus on the \emph{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., $\sum_{v\in V} d(v,C)$), 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.~\cite{eube2025esa} 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 $O(\log^2 n)$-approximation. For the general version, we develop a bicriteria framework that opens $O(k\log n)$ centers while achieving an $O(\log^2 n)$-approximation in cost. %Our results provide a different LP-based approach for handling connectivity constraints in clustering problems and demonstrate that configuration LPs, covering LPs, and rooted density oracles can be combined effectively to obtain approximation guarantees for clustering objectives under graph-theoretic constraints.
CommentsThis paper has been accepted in Approx 2026
