On the Complexity of Minimum Riesz s-Energy Subset Selection in Euclidean and Ultrametric Spaces

Abstract

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.