We prove that, for every fixed $θ_0>0$, selecting a subset of prescribed cardinality that maximizes the Solow--Polasky diversity indicator is NP-hard for finite point sets in $\mathbb{R}^2$ with the Euclidean metric, and therefore also for finite point sets in $\mathbb{R}^d$ for every fixed dimension $d \ge 2$. This strictly strengthens our earlier NP-hardness result for general metric spaces by showing that hardness persists under the severe geometric restriction to the Euclidean plane. At the same time, the Euclidean proof technique is different from the conceptually easier earlier argument for arbitrary metric spaces, and that general metric-space construction does not directly translate to the Euclidean setting. In the earlier proof one can use an exact construction tailored to arbitrary metrics, essentially exploiting a two-distance structure. In contrast, such an exact realization is unavailable in fixed-dimensional Euclidean space, so the present reduction requires a genuinely geometric argument. Our Euclidean proof is based on two distance thresholds, which allow us to separate yes-instances from no-instances by robust inequalities rather than by the exact construction used in the general metric setting. The main technical ingredient is a bounded-box comparison lemma for the nonlinear objective $\mathbf{1}^{\top}Z^{-1}\mathbf{1}$, where $Z_{ij}=e^{-θ_0 d(x_i,x_j)}$. This lemma controls the effect of perturbations in the pairwise distances well enough to transfer the gap created by the reduction. The reduction is from \emph{Geometric Unit-Disk Independent Set}. We present the main argument in geometric form for finite subsets of $\mathbb{R}^2$, with an appendix supplying the bit-complexity details needed for polynomial-time reducibility.

翻译：我们证明，对于任意固定的$θ_0>0$，在具有欧几里得度量的$\mathbb{R}^2$有限点集中，选择指定基数且最大化Solow-Polasky多样性指标的子集是NP难的，因此对于$\mathbb{R}^d$中任意固定维度$d \ge 2$的有限点集也是如此。这严格加强了我们先前的关于一般度量空间的NP难结果，表明在欧几里得平面的严重几何限制下，该困难性依然存在。同时，欧几里得证明技术不同于先前针对任意度量空间的概念上更简单的论证，而那个一般的度量空间构造方法不能直接转化为欧几里得场景。在之前的证明中，我们可以使用针对任意度量定制的精确构造，本质上利用了“两距离”结构。相比之下，这样的精确实现在固定维度的欧几里得空间中是不可行的，因此目前的归约需要真正几何化的论证。我们的欧几里得证明基于两个距离阈值，使我们能够通过稳健不等式而非一般度量设置中使用的精确构造，将肯定实例与否定实例区分开来。主要技术成分是针对非线性目标函数$\mathbf{1}^{\top}Z^{-1}\mathbf{1}$（其中$Z_{ij}=e^{-θ_0 d(x_i,x_j)}$）的有界盒比较引理。该引理足够精细地控制成对距离扰动的影响，从而传递归约所产生的间隙。该归约来自\emph{几何单位圆盘独立集}问题。我们以几何形式给出$\mathbb{R}^2$有限子集的主要论证，附录中提供了多项式时间可归约性所需的位复杂性细节。