Riesz Energy Subset Selection in the Euclidean Plane is NP-Hard: A Reduction from the Ising Model on Planar Cubic Graphs

[2608.23506] Riesz Energy Subset Selection in the Euclidean Plane is NP-Hard: A Reduction from the Ising Model on Planar Cubic Graphs Skip to main content Search Submit Donate Log in Computer Science > Computational Geometry arXiv:2608.23506 (cs) [Submitted on 24 Aug 2026 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: Riesz Energy Subset Selection in the Euclidean Plane is NP-Hard: A Reduction from the Ising Model on Planar Cubic Graphs Authors: Michael Emmerich View PDF 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 $45^\circ$ 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. Comments: Keywords: Riesz energy, subset selection, NP-completeness, geometric optimization, planar cubic graphs, independent set, Ising model, computational geometry Update V3.0: Adversarial proof audit with Claude Fable 5 and augmented Git with lean proof support and audit result. Sharpened some constants. No fundamental changes or new results w.r.t. previous versions Subjects: Computational Geometry (cs.CG) ; Computational Complexity (cs.CC); Computational Physics (physics.comp-ph) MSC classes: 68Q17 Cite as: arXiv:2608.23506 [cs.CG] (or arXiv:2608.23506v3 [cs.CG] for this version) https://doi.org/10.48550/arXiv.2608.23506 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Michael Emmerich [ view email ] [v1] Mon, 24 Aug 2026 17:09:26 UTC (32 KB) [v2] Tue, 25 Aug 2026 07:34:06 UTC (32 KB) [v3] Thu, 27 Aug 2026 07:24:36 UTC (37 KB) Full-text links: Access Paper: View PDF TeX Source view license Additional Features Audio Summary Current browse context: cs.CG < prev | next > new | recent | 2026-08 Change to browse by: cs cs.CC physics physics.comp-ph References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

Related documents

• Comparing the Lee-Yang gap with the mass gap in the sub-critical planar Ising model#1051841
• Sharp Edge-Edit Bounds at Every Level for Leaky Positive Semidefinite Forcing#1037487
• Tangential Surface Pressures and Boundary Localization in Disordered Edwards--Anderson Models#1034034
• Parity-selected Bethe-Salpeter equation: symmetry-adapted regularization of the exciton-phonon self-energy in polar semiconductors#1117621
• The Ramsey threshold for trees versus odd cycles#974606