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The aim in packing problems is to decide if a given set of pieces can be placed inside a given container.
“A Universality Theorem for Nested Polytopes” Preprint., 2019
Michael Dobbins, Andreas Holmsen and Tillmann Miltzow · 1908
Earlier work this paper cites.
“Optimal Curve Straightening is ∃ ℝ \exists\mathbb{R} -Complete” Preprint, 2019
Jeff Erickson · 1908
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“The Complexity of Positive Semidefinite Matrix Factorization”
Yaroslav Shitov · 1909
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“Über die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene”
A. Thue · 1910
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“Über die dichteste Kugellagerung”
László Tóth · 1942
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“On packing squares with equal squares”
Paul Erdős and Ron Graham · 1975
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“Optimal packing and covering in the plane are NP-complete”
Robert. Fowler, Michael. Paterson and Steven. Tanimoto · 1981
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“Some algebraic and geometric computations in PSPACE”
John Canny · 1988
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“The universality theorems on the classification problem of configuration varieties and convex polytopes varieties”
Nicolai Mnëv · 1988
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“Generalized Planar Matching”
Francine Berman, David. Johnson, Frank Leighton, Peter. Shor and Larry Snyder · 1990
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“Packing squares into a square”
Joseph Leung, Tommy Tam, Chin Wong, Gilbert Young and Francis Chin · 1990
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“Some provably hard crossing number problems”
Daniel Bienstock · 1991
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“Stretchability of pseudolines is NP-hard”
Peter. Shor · 1991
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“Cutting and packing in production and distribution: A typology and bibliography”, 1992
Harald Dyckhoff and Ute Finke · 1992
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“Cutting and packing problems: a categorized, application-orientated research bibliography”
Paul Sweeney and Elizabeth Paternoster · 1992
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“Realization spaces of 4-polytopes are universal”
Jürgen Richter-Gebert and Günter. Ziegler · 1995
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“Translational Polygon Containment and Minimal Enclosure using Linear Programming Based Restriction”
Victor Milenkovic · 1996
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“Multiple Translational Containment. Part I: An Approximate Algorithm”
Karen. Daniels and Victor Milenkovic · 1997
Earlier work this paper cites.
“Multiple Translational Containment. Part II: Exact Algorithms”
Victor Milenkovic · 1997
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“Translational polygon containment and minimal enclosure using mathematical programming”
Karen. Daniels and Victor Milenkovic · 1999
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“Rotational polygon containment and minimum enclosure using only robust 2D constructions”
Victor Milenkovic · 1999
Earlier work this paper cites.
“A review of the application of meta-heuristic algorithms to 2D strip packing problems”
Eva Hopper and Brian Turton · 2001
Cited alongside, same era.
“A rational quartic Bézier representation for conics”
Lian Fang · 2002
Cited alongside, same era.
“Improved Dense Packings of Congruent Squares in a Square”
Thierry Gensane and Philippe Ryckelynck · 2005
Cited alongside, same era.
“A proof of the Kepler conjecture”
Thomas. Hales · 2005
Cited alongside, same era.
“Algorithms in real algebraic geometry”
Saugata Basu, Richard Pollack and Marie-Françoise Roy · 2006
Cited alongside, same era.
“The geometry of nesting problems: A tutorial”
Julia Bennell and Jose Oliveira · 2006
Cited alongside, same era.
“Fixed Points, Nash Equilibria, and the Existential Theory of the Reals”
Marcus Schaefer and Daniel Štefankovič · 2017
Later among the works it cites.
“The sphere packing problem in dimension 8”
Maryna. Viazovska · 2017
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“Intersection Graphs of Rays and Grounded Segments”
Jean Cardinal, Stefan Felsner, Tillmann Miltzow, Casey Tompkins and Birgit Vogtenhuber · 2018
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“ ∃ ℝ \exists\mathbb{R} -Completeness for Decision Versions of Multi-Player (Symmetric) Nash Equilibria”
Jugal Garg, Ruta Mehta, Vijay. Vazirani and Sadra Yazdanbod · 2018
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“Planar graphs and faces areas – Area-Universality” PhD thesis, Technische Universität Berlin, 2018
Linda Kleist · 2018
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“Segment representations with small resolution”
Therese Biedl · 2019
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“A tutorial in irregular shape packing problems”
Julia Bennell and Jose Oliveira · 2008
Cited alongside, same era.
“Complexity of some geometric and topological problems”
Marcus Schaefer · 2009
Cited alongside, same era.
“Circle packing for origami design is hard” Preprint, 2010
Erik. Demaine, Sándor. Fekete and Robert. Lang · 2010
Cited alongside, same era.
“Packing identical simple polygons is NP-hard” Preprint, 2012
Sarah. Allen and John Iacono · 2012
Cited alongside, same era.
“Sphere and Dot Product Representations of Graphs”
Ross. Kang and Tobias Müller · 2012
Cited alongside, same era.
“Integer realizations of disk and segment graphs”
Colin McDiarmid and Tobias Müller · 2012
Cited alongside, same era.
Later among the works it cites.
“Irregular packing problems: A review of mathematical models”
Aline.S. Leao, Franklina.B. Toledo, José Oliveira, Mariaónia Carravilla and Ramón Alvarez-Valdés · 2019
Later among the works it cites.
“Framework for ∃ ℝ \exists\mathbb{R} -Completeness of Two-Dimensional Packing Problems”
Mikkel Abrahamsen, Tillmann Miltzow and Nadja Seiferth · 2020
Closest in time.
“Efficient Packings of Unit Squares in a Large Square”
Fan Chung and Ron Graham · 2020
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“Smoothing the gap between NP and ER”
Jeff Erickson, Ivor van der Hoog and Tillmann Miltzow · 2020
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“On the Two-Dimensional Knapsack Problem for Convex Polygons”
Arturo. Merino and Andreas Wiese · 2020
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“Covering Polygons is Even Harder”
Mikkel Abrahamsen · 2021
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“The art gallery problem is ∃ ℝ \exists\mathbb{R} -complete”
Mikkel Abrahamsen, Anna Adamaszek and Tillmann Miltzow · 2021
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“Training Neural Networks is ∃ ℝ \exists\mathbb{R} -complete”
Mikkel Abrahamsen, Linda Kleist and Tillmann Miltzow · 2021
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“On the Computational Complexity of Decision Problems About Multi-player Nash Equilibria”
Marieølbøll Berthelsen and Kristoffer Hansen · 2022
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“The Complexity of Drawing a Graph in a Polygonal Region”
Anna Lubiw, Tillmann Miltzow and Debajyoti Mondal · 2022
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“Geometric Embeddability of Complexes Is ∃ ℝ \exists\mathbb{R} -Complete”
Mikkel Abrahamsen, Linda Kleist and Tillmann Miltzow · 2023
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“Training Fully Connected Neural Networks is ∃ ℝ \exists\mathbb{R} -Complete”
Daniel Bertschinger, Christoph Hertrich, Paul Jungeblut, Tillmann Miltzow and Simon Weber · 2023
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“Completeness for the Complexity Class ∀ ∃ ℝ \forall\exists\mathbb{R} and Area-Universality”
Michael Dobbins, Linda Kleist, Tillmann Miltzow and Pawel Rzazewski · 2023
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“On Classifying Continuous Constraint Satisfaction problems”
Tillmann Miltzow and Reinier. Schmiermann · 2024
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