Riesz Energy Minimization Facility Location Problem in the Plane: Complexity and a Polynomial-Time Approximation Scheme

• Defines the Riesz Energy Minimization Facility Location Problem • Proves NP-hardness in general metrics via reduction from Independent Set • Analyzes limiting behavior interpolating dispersion and uniform selection • Develops a PTAS for the planar case under minimum-separation condition • Achieves ( 1 + ϵ ) approximation in polynomial time for fixed facility count. This article introduces a geometric facility location problem that minimizes total Riesz interaction energy among p selected sites in the Euclidean plane. The objective generalizes the classical p -dispersion criterion by penalizing spatial proximity through inverse power distances, thereby promoting global dispersion in a smooth manner. The problem is shown to be NP-hard in general metric spaces. A Polynomial-Time Approximation Scheme (PTAS) is developed for the planar case for a truncated Riesz objective (for fixed p ), with the same approximation guarantee extending to the original objective whenever the candidate set satisfies a natural minimum-separation condition. Furthermore, limiting analysis shows that the Riesz objective provides a smooth interpolation between strict dispersion and uniform selection, depending on the strength of the interaction exponent.