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

TL;DR

This paper investigates the computational complexity of selecting well-separated subsets based on Riesz s-energy in different metric spaces, revealing NP-hardness in Euclidean spaces and tractability in ultrametric spaces.

Contribution

It proves NP-hardness of the problem in Euclidean spaces for variable s and provides an efficient dynamic programming solution for ultrametric spaces, clarifying the complexity landscape.

Findings

NP-hardness persists for Euclidean point sets when s varies.

Ultrametric spaces allow polynomial-time exact solutions via dynamic programming.

Ordered one-dimensional Euclidean case admits simple dynamic programming for MPD, but not for Riesz energy.

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.