The densest hemisphere problem
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Exact computation of the Tukey halfspace median is NP-hard for d ≥ 3.
“whose exact computation is NP-hard for d ≥ 3 (Johnson and Preparata, 1978;Aloupis, 2006)”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Johnson and Preparata, 1978
Section 2 establishes NP-completeness for the hemisphere problems when the dimension is not fixed in advance. The paper does not establish NP-hardness for computing the Tukey halfspace median or for any fixed dimension d ≥ 3; Sections 3–4 instead give polynomial-time algorithms for the hemisphere problems at fixed d. (LLM, conf=0.84)
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Research works printed but not linked: 1
- [2]2. A. V. Aho, J. E. Hopcroft and J. D. Ullman, Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, Mass., 1974. (Chapter 10)
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Support Vector Machines with the Ramp Loss and the Hard Margin Loss
Brooks, J. P. (2011) · Operations Research · doi:10.1287/opre.1100.0854 · Cited by 145 · see citing works
RoPoLL: Robust Panel of LLM Judges
Acharya, A., Pan, K. W., & Verkhovsky, B. (2026) · arXiv · doi:10.48550/arxiv.2606.30931
- Authors
- Johnson, D. S., & Preparata, F. P.
- Affiliations
- University of Illinois Urbana-Champaign (US)
- Venue
- Theoretical Computer Science
- Publisher
- Elsevier BV
- Published
- 1978
- Volume
- 6
- Issue
- 1
- Pages
- 93-107
- ISSN
- 0304-3975 (ISSN-L)
- OpenAlex
- W2053113791
- Cited by
- 152
- References
- 0
- Citations by year
- 2025: 6 · 2024: 2 · 2023: 4 · 2022: 2 · 2021: 12 · 2020: 6 · 2019: 9 · 2018: 7 · 2017: 6 · 2016: 7 · 2015: 6 · 2014: 2 · 2013: 6 · 2012: 3
- FWCI
- 0.53
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Theoretical Computer Science (publishedVersion)
- Illinois Digital Environment for Access to Learning and Scholarship (University of Illinois at Urbana-Champaign) (submittedVersion)
- Defense Technical Information Center (DTIC) (submittedVersion)
- Funders
- National Science Foundation
- U.S. Air Force (C-0259, DAAB-07-72, DAAB-07-72-C-0259)
- U.S. Navy
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Complexity and Algorithms in Graphs
- Physical Sciences › Computer Science › Computer Graphics and Computer-Aided Design › Computational Geometry and Mesh Generation
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Advanced Combinatorial Mathematics
- Keywords
- unit sphere, NP-completeness, feasibility problem, polynomial-time algorithms, fixed dimension, computational geometry
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