The Solow--Polasky diversity indicator (or magnitude) is a classical measure of diversity based on pairwise distances. It has applications in ecology, conservation planning, and, more recently, in algorithmic subset selection and diversity optimization. In this note, we investigate the computational complexity of selecting a subset of fixed cardinality from a finite set so as to maximize the Solow--Polasky diversity value. We prove that this problem is NP-hard in general metric spaces. The reduction is from the classical Independent Set problem and uses a simple metric construction containing only two non-zero distance values. Importantly, the hardness result holds for every fixed kernel parameter $θ_0>0$; equivalently, by rescaling the metric, one may fix the parameter to $1$ without loss of generality. A central point is that this is not a boilerplate reduction: because the Solow--Polasky objective is defined through matrix inversion, it is a nontrivial nonlinear function of the distances. Accordingly, the proof requires a dedicated strict-monotonicity argument for the specific family of distance matrices arising in the reduction; this strict monotonicity is established here for that family, but it is not assumed to hold in full generality. We also explain how the proof connects to continuity and monotonicity considerations for diversity indicators.

翻译：索洛-波拉斯基多样性指标（或称“幅度”）是一种基于成对距离的经典多样性度量方法。该指标广泛应用于生态学、保护规划领域，并在近期被拓展至算法子集选择与多样性优化问题中。本文研究了从有限集合中选择固定基数子集以最大化索洛-波拉斯基多样性值的计算复杂度问题，证明该问题在一般度量空间中是NP难的。该难度结论通过从经典独立集问题的归约得到，并采用仅含两种非零距离值的简单度量构造。重要的是，该难度结果对所有固定核参数$θ_0>0$成立；等价地，通过度量重缩放，可将参数固定为$1$而不失一般性。核心要点在于，此归约并非标准模板式归约：由于索洛-波拉斯基目标函数通过矩阵求逆定义，它是距离的非平凡非线性函数。因此，证明需要针对归约中出现的特定距离矩阵族建立专用的严格单调性论证——本文针对该族矩阵建立了此类严格单调性，但并非假定其在全一般性下成立。最后，我们阐释该证明如何与多样性指标的连续性和单调性考量相关联。