Riesz Energy Subset Selection in the Euclidean Plane is NP-Hard: A Reduction from the Ising Model on Planar Cubic Graphs
Michael Emmerich
Abstract
We prove that minimum Riesz s-energy subset selection in the Euclidean plane is NP-complete already for the fixed exponent s=2. To our knowledge, this is the first Euclidean hardness result for exact Riesz-energy subset selection in which both the ambient dimension and the exponent are fixed. The reduction uses Barahona's planar cubic Ising model with uniform field. A spin is encoded by one diagonal of a four-point square. Axis-aligned selector chains implement ferromagnetic consistency, while a 45 terminal geometry yields an antiferromagnetic source interaction. Rational diagonal perturbations realize the magnetic field, and all remaining interactions are dominated by polynomial separation. Because s=2 and all coordinates are rational, every constructed energy and the decision threshold are rational exactly.
Create a lesson
Related papers
Lunar Generalizations of the Euclidean Minimum Spanning Tree in the Plane and their Expected Costs
Ondřej Draganov, Herbert Edelsbrunner, Sophie Rosenmeier et al.
Quadratic Complexity of Voronoi Diagrams in R3 for Lines in a Single Ruling of a Regulus
Eunku Park
Sequential Euclidean tree construction with exponential memory: distributional performance and worst-case guarantees
Pedro M. M. de Castro
Computing an e-net of a closed hyperbolic surface
Vincent Delecroix, Vincent Despré, Camille Lanuel et al.
On Angle-optimization and Simplification of Degree-1 Homology Representatives
Emerson G. Escolar, Yuta Shimada
A fast improved quasi-physical dynamic algorithm for efficient wireless coverage in convex polygonal regions
Zeping Yi, Yongjun Wanga, Baoshan Wang et al.