Maximum Solow--Polasky Diversity Subset Selection Is NP-hard Even in the Euclidean Plane

Abstract

We prove that, for every fixed $\theta_0>0$, 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 $Z_{ij}=e^{-\theta_0 d(x_i,x_j)}$. 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.