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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

The classic Ham-Sandwich theorem states that for any d measurable sets in ℝ^d, there is a hyperplane that bisects them simultaneously. An extension by Bárány, Hubard, and Jerónimo [DCG 2008] states that if the sets are convex and well-separated, then for any given α₁, … , α_d ∈ [0, 1], there is a unique oriented hyperplane that cuts off a respective fraction α₁, … , α_d from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the α-Ham-Sandwich theorem. They gave an algorithm to find the hyperplane in time O(n (log n)^{d-3}), where n is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]).
Recently, Fearnley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called Unique End-of-Potential Line (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the α-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the P-matrix linear complementarity problem.

Man-Kwun Chiu, Aruni Choudhary, and Wolfgang Mulzer. Computational Complexity of the α-Ham-Sandwich Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 31:1-31:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chiu_et_al:LIPIcs.ICALP.2020.31, author = {Chiu, Man-Kwun and Choudhary, Aruni and Mulzer, Wolfgang}, title = {{Computational Complexity of the \alpha-Ham-Sandwich Problem}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {31:1--31:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.31}, URN = {urn:nbn:de:0030-drops-124382}, doi = {10.4230/LIPIcs.ICALP.2020.31}, annote = {Keywords: Ham-Sandwich Theorem, Computational Complexity, Continuous Local Search} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Tverberg’s theorem states that for any k ≥ 2 and any set P ⊂ ℝ^d of at least (d + 1)(k - 1) + 1 points, we can partition P into k subsets whose convex hulls have a non-empty intersection. The associated search problem lies in the complexity class PPAD ∩ PLS, but no hardness results are known. In the colorful Tverberg theorem, the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Bárány, and Mustafa [SODA 2019] gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of P. The argument is constructive, but does not result in a polynomial-time algorithm.
We present a deterministic algorithm that finds for any n-point set P ⊂ ℝ^d and any k ∈ {2, … , n} in O(nd ⌈log k⌉) time a k-partition of P such that there is a ball of radius O((k/√n)diam(P)) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, and that there are no approximation algorithms that are truly polynomial in any dimension, our result provides a remarkably efficient and simple new notion of approximation.
Our main contribution is to generalize Sarkaria’s method [Israel Journal Math., 1992] to reduce the Tverberg problem to the Colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem.

Aruni Choudhary and Wolfgang Mulzer. No-Dimensional Tverberg Theorems and Algorithms. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 31:1-31:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{choudhary_et_al:LIPIcs.SoCG.2020.31, author = {Choudhary, Aruni and Mulzer, Wolfgang}, title = {{No-Dimensional Tverberg Theorems and Algorithms}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {31:1--31:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.31}, URN = {urn:nbn:de:0030-drops-121893}, doi = {10.4230/LIPIcs.SoCG.2020.31}, annote = {Keywords: Tverberg’s theorem, Colorful Carath\'{e}odory Theorem, Tensor lifting} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes constitutes an expensive task because of a combinatorial explosion in the complex size. For n points in R^d, we present a scheme to construct a 4.24-approximation of the multi-scale filtration of the Rips complex in the L-infinity metric, which extends to a O(d^{0.25})-approximation of the Rips filtration for the Euclidean case. The k-skeleton of the resulting approximation has a total size of n2^{O(d log k)}. The scheme is based on the integer lattice and on the barycentric subdivision of the d-cube.

Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Improved Approximate Rips Filtrations with Shifted Integer Lattices. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 28:1-28:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{choudhary_et_al:LIPIcs.ESA.2017.28, author = {Choudhary, Aruni and Kerber, Michael and Raghvendra, Sharath}, title = {{Improved Approximate Rips Filtrations with Shifted Integer Lattices}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {28:1--28:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.28}, URN = {urn:nbn:de:0030-drops-78259}, doi = {10.4230/LIPIcs.ESA.2017.28}, annote = {Keywords: Persistent homology, Rips filtrations, Approximation algorithms, Topological Data Analysis} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in R^d, we obtain a O(d)-approximation with at most n2^{O(d log k)} simplices of dimension k or lower. In conjunction with dimension reduction techniques, our approach yields a O(polylog (n))-approximation of size n^{O(1)} for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry.
Building on the same geometric concept, we also present a lower bound result on the size of an approximate filtration: we construct a point set for which every (1+epsilon)-approximation of the Cech filtration has to contain n^{Omega(log log n)} features, provided that epsilon < 1/(log^{1+c}n) for c in (0,1).

Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Polynomial-Sized Topological Approximations Using the Permutahedron. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 31:1-31:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{choudhary_et_al:LIPIcs.SoCG.2016.31, author = {Choudhary, Aruni and Kerber, Michael and Raghvendra, Sharath}, title = {{Polynomial-Sized Topological Approximations Using the Permutahedron}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {31:1--31:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.31}, URN = {urn:nbn:de:0030-drops-59236}, doi = {10.4230/LIPIcs.SoCG.2016.31}, annote = {Keywords: Persistent Homology, Topological Data Analysis, Simplicial Approximation, Permutahedron, Approximation Algorithms} }

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