同时在所有对称单调范数下的近似最小树覆盖
Approximate Minimum Tree Cover in All Symmetric Monotone Norms Simultaneously
AI总结：
本文研究以最小生成树权值为簇代价的聚类问题，改造 Min-Max Tree Cover 算法，得到可同时近似所有单调对称范数的纯组合多项式时间算法，并扩展到含 depot 约束情形，同时证明单一 $\ell_p$ 目标下已为 APX-hard。
摘要：
我们研究将度量空间中的一组 $n$ 个对象划分为 $k$ 个簇 $V_1,\dots,V_k$ 的问题。聚类质量通过考察簇代价向量，并最小化该向量的某种单调对称范数来衡量（特别地，这包括 $\ell_p$ 范数）。对于各簇的代价，我们取 $V_i$ 中对象上最小权生成树的权值；它既可以代表遍历该簇中所有对象的代价，也可以作为与 Single-Linkage Clustering（单链聚类）类似、对形状不变的簇密度度量。\nEven、Garg、Könemann、Ravi、Sinha（Oper. Res. Lett.，2004）曾研究这一设定中最小化最大簇权值的情形（即使用 $\ell_\infty$），称之为 Min-Max Tree Cover，并给出了常数因子近似。我们细致地改造他们的算法，以计算相对于所有单调对称范数同时近似最优的解，并说明如何在多项式时间内找到这些解。事实上，我们的算法是纯组合式的，能够在不到一秒内处理含 10,000 个点的度量空间。\n作为扩展，我们还考虑另一种情形：给定的不是目标簇数，而是空间中的一组 depot（站点），并且每个簇都应至少包含一个这样的站点。对于该设定，我们同样能够给出一个多项式时间算法，同时针对所有单调对称范数计算常数因子近似。\n为表明这些算法结果在可达到的精确近似常数意义下是紧的，我们还证明：即使目标只考虑单一的一个 $\ell_p$ 范数，此类聚类问题也已经是 APX-hard 的。
英文摘要：
We study the problem of partitioning a set of $n$ objects in a metric space into $k$ clusters $V_1,\dots,V_k$. The quality of the clustering is measured by considering the vector of cluster costs and then minimizing some monotone symmetric norm of that vector (in particular, this includes the $\ell_p$-norms). For the costs of the clusters we take the weight of a minimum-weight spanning tree on the objects in~$V_i$, which may serve as a proxy for the cost of traversing all objects in the cluster, but also as a shape-invariant measure of cluster density similar to Single-Linkage Clustering. This setting has been studied by Even, Garg, Könemann, Ravi, Sinha (Oper. Res. Lett.}, 2004) for the setting of minimizing the weight of the largest cluster (i.e., using $\ell_\infty$) as Min-Max Tree Cover, for which they gave a constant-factor approximation. We provide a careful adaptation of their algorithm to compute solutions which are approximately optimal with respect to all monotone symmetric norms simultaneously, and show how to find them in polynomial time. In fact, our algorithm is purely combinatorial and can process metric spaces with 10,000 points in less than a second. As an extension, we also consider the case where instead of a target number of clusters we are provided with a set of depots in the space such that every cluster should contain at least one such depot. For this setting also we are able to give a polynomial time algorithm computing a constant factor approximation with respect to all monotone symmetric norms simultaneously. To show that the algorithmic results are tight up to the precise constant of approximation attainable, we also prove that such clustering problems are already APX-hard when considering only one single $\ell_p$ norm for the objective.
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