We study the computational complexity of exact cardinality-constrained minimum Riesz $s$-energy subset selection in finite metric spaces: given $n$ points, select $k<n$ points of minimum Riesz $s$-energy. The objective sums inverse-power pair interactions and therefore promotes well-separated subsets; as $s$ becomes large, it increasingly approaches a bottleneck criterion governed by the closest selected pair, linking it to minimum pairwise distance (MPD). Building on the general-metric NP-hardness result of Pereverdieva et al. (2025), we prove that NP-hardness persists for point sets in the Euclidean plane when $s$ is part of the input. In contrast, finite ultrametric spaces form an exact tractable regime: on rooted binary ultrametric trees with $n$ leaves, an optimal size-$k$ subset can be computed by dynamic programming in $O(nk^2)$ time. We also discuss the ordered one-dimensional Euclidean case, where the classical MPD objective admits simple dynamic programming, but the additive Riesz energy does not appear to allow the same state compression. Finally, we explain why one natural route to fixed-$s$ Euclidean hardness does not close: Fowler-style 3SAT gadgets, together with zeta-function bounds for far-field interactions, show why this approach still requires an exponent depending on $k$. Together, these results provide a compact complexity landscape for a natural diversity or dispersion objective, distinguishing Euclidean hardness, ultrametric tractability, and the ordered one-dimensional case.

翻译：我们研究了有限度量空间中精确基数约束的最小Riesz $s$-能量子集选择问题的计算复杂性：给定$n$个点，选择具有最小Riesz $s$-能量的$k<n$个点。该目标函数对逆幂对相互作用求和，因此促进良好分离的子集；随着$s$增大，它逐渐趋近于由最近选择对决定的瓶颈准则，从而与最小成对距离问题相关联。基于Pereverdieva等人(2025)的一般度量NP困难性结果，我们证明了当$s$作为输入的一部分时，欧几里得平面中点集的该问题仍保持NP困难性。相比之下，有限超度量空间构成精确可解情形：在具有$n$个叶子的有根二叉超度量树上，可以通过动态规划在$O(nk^2)$时间内计算出最优大小为$k$的子集。我们还讨论了一维有序欧几里得情形，其中经典的最小成对距离目标具有简单动态规划，但可加Riesz能量似乎不允许相同的状态压缩。最后，我们解释了为何固定$s$欧几里得困难性的一条自然路径无法实现：Fowler式3SAT小工具结合远场相互作用的zeta函数界，表明该方法仍需要依赖于$k$的指数。这些结果共同为这一自然的多样性或离散度目标提供了紧凑的复杂性景观，区分了欧几里得困难性、超度量可解性以及一维有序情形。