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1. Cutting Polygons into Small Pieces with Chords: Laser-Based Localization

   https://doi.org/10.4230/LIPIcs.ESA.2020.7  

   Arkin, Esther; Das, Rathish; Gao, Jie; Goswami, Mayank; Mitchell, Joseph; Polishchuk, Valentin; Toth, Csaba. (August 2020, Leibniz international proceedings in informatics)

   Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem. 

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2. Polygons with Prescribed Angles in 2D and 3D

   https://doi.org/10.1007/978-3-030-68766-3_11  

   Efrat, Alon; Fulek, Radoslav; Kobourov, Stephen; Toth, Csaba D. (February 2021, Graph Drawing and Network Visualization)

   null (Ed.)

   We consider the construction of a polygon P with n vertices whose turning angles at the vertices are given by a sequence A=(α0,…,αn−1) , αi∈(−π,π) , for i∈{0,…,n−1} . The problem of realizing A by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an angle graph. In 2D, we characterize sequences A for which every generic polygon P⊂R2 realizing A has at least c crossings, for every c∈N , and describe an efficient algorithm that constructs, for a given sequence A, a generic polygon P⊂R2 that realizes A with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence A can be realized by a (not necessarily generic) polygon P⊂R3 , and for every realizable sequence the algorithm finds a realization. 

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3. Polygons with Prescribed Angles in 2D and 3D

   Efrat, A; Fulek, R; Kobourov, S; Toth, C (October 2020, 28th International Symposium on Graph Drawing and Network Visualization (GD))

   We consider the construction of a polygon P with n vertices whose turning angles at the vertices are given by a sequence A = (α0 , . . . , αn−1 ), αi ∈ (−π,π), for i ∈ {0,...,n − 1}. The problem of realizing A by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an angle graph. In 2D, we characterize sequences A for which every generic polygon P ⊂ R2 realizing A has at least c crossings, and describe an efficient algorithm that constructs, for a given sequence A, a generic polygon P ⊂ R2 that realizes A with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence A can be realized by a (not necessarily generic) polygon P ⊂ R3, and for every realizable sequence finds a realization. 

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4. Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes

   https://doi.org/10.2140/paa.2025.7.927  

   Liu, Jian-Guo; Pego, Robert L (January 2025, Pure and Applied Analysis)

   We study all the ways that a given convex body in d dimensions can break into countably many pieces that move away from each other rigidly at constant velocity, with no rotation or shearing. The initial velocity field is locally constant a.e., but may be continuous and/or fail to be integrable. For any choice of mass-velocity pairs for the pieces, such a motion can be generated by the gradient of a convex potential that is affine on each piece. We classify such potentials in terms of a countable version of a theorem of Alexandrov for convex polytopes, and prove a stability theorem. For bounded velocities, there is a bijection between the mass-velocity data and optimal transport flows (Wasserstein geodesics) that are locally incompressible. Given any rigidly breaking velocity field that is the gradient of a continuous potential, the convexity of the potential is established under any of several conditions, such as the velocity field being continuous, the potential being semiconvex, the mass measure generated by a convexified transport potential being absolutely continuous, or there being a finite number of pieces. Also we describe a number of curious and paradoxical examples having fractal structure. 

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5. Maximal Simplification of Polyhedral Reductions

   https://doi.org/10.1145/3704839  

   Narmour, Louis; Yuki, Tomofumi; Rajopadhye, Sanjay (January 2025, Proceedings of the ACM on Programming Languages)

   Reductions combine collections of input values with an associative and often commutative operator to produce collections of results. When the same input value contributes to multiple outputs, there is an opportunity to reuse partial results, enabling reduction simplification. Simplification often produces a program with lower asymptotic complexity. Typical compiler optimizations yield, at best, a constant fold speedup, but a complexity improvement from, say, cubic to quadratic complexity yields unbounded speedup for sufficiently large problems. It is well known that reductions in polyhedral programs may be simplified automatically, but previous methods cannot exploit all available reuse. This paper resolves this long-standing open problem, thereby attaining minimal asymptotic complexity in the simplified program. We propose extensions to prior work on simplification to support any independent commutative reduction. At the heart of our approach is piece-wise simplification, the notion that we can split an arbitrary reduction into pieces and then independently simplify each piece. However, the difficulty of using such piece-wise transformations is that they typically involve an infinite number of choices. We give constructive proofs to deal with this and select a finite number of pieces for simplification. 

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