4 Search Results for "Meunier, Frédéric"


Document
No-Dimensional Tverberg Theorems and Algorithms

Authors: Aruni Choudhary and Wolfgang Mulzer

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


Abstract
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.

Cite as

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}
}
Document
Approximating Hit Rate Curves using Streaming Algorithms

Authors: Zachary Drudi, Nicholas J. A. Harvey, Stephen Ingram, Andrew Warfield, and Jake Wires

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
A hit rate curve is a function that maps cache size to the proportion of requests that can be served from the cache. (The caching policy and sequence of requests are assumed to be fixed.) Hit rate curves have been studied for decades in the operating system, database and computer architecture communities. They are useful tools for designing appropriate cache sizes, dynamically allocating memory between competing caches, and for summarizing locality properties of the request sequence. In this paper we focus on the widely-used LRU caching policy. Computing hit rate curves is very efficient from a runtime standpoint, but existing algorithms are not efficient in their space usage. For a stream of m requests for n cacheable objects, all existing algorithms that provably compute the hit rate curve use space linear in n. In the context of modern storage systems, n can easily be in the billions or trillions, so the space usage of these algorithms makes them impractical. We present the first algorithm for provably approximating hit rate curves for the LRU policy with sublinear space. Our algorithm uses O( p^2 * log(n) * log^2(m) / epsilon^2 ) bits of space and approximates the hit rate curve at p uniformly-spaced points to within additive error epsilon. This is not far from optimal. Any single-pass algorithm with the same guarantees must use Omega(p^2 + epsilon^{-2} + log(n)) bits of space. Furthermore, our use of additive error is necessary. Any single-pass algorithm achieving multiplicative error requires Omega(n) bits of space.

Cite as

Zachary Drudi, Nicholas J. A. Harvey, Stephen Ingram, Andrew Warfield, and Jake Wires. Approximating Hit Rate Curves using Streaming Algorithms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 225-241, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{drudi_et_al:LIPIcs.APPROX-RANDOM.2015.225,
  author =	{Drudi, Zachary and Harvey, Nicholas J. A. and Ingram, Stephen and Warfield, Andrew and Wires, Jake},
  title =	{{Approximating Hit Rate Curves using Streaming Algorithms}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{225--241},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.225},
  URN =		{urn:nbn:de:0030-drops-53056},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.225},
  annote =	{Keywords: Cache analysis, hit rate curves, miss rate curves, streaming algorithms}
}
Document
Computational Aspects of the Colorful Carathéodory Theorem

Authors: Wolfgang Mulzer and Yannik Stein

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)


Abstract
Let P_1,...,P_{d+1} be d-dimensional point sets such that the convex hull of each P_i contains the origin. We call the sets P_i color classes, and we think of the points in P_i as having color i. A colorful choice is a set with at most one point of each color. The colorful Caratheodory theorem guarantees the existence of a colorful choice whose convex hull contains the origin. So far, the computational complexity of finding such a colorful choice is unknown. We approach this problem from two directions. First, we consider approximation algorithms: an m-colorful choice is a set that contains at most m points from each color class. We show that for any fixed epsilon > 0, an (epsilon d)-colorful choice containing the origin in its convex hull can be found in polynomial time. This notion of approximation has not been studied before, and it is motivated through the applications of the colorful Caratheodory theorem in the literature. In the second part, we present a natural generalization of the colorful Caratheodory problem: in the Nearest Colorful Polytope problem (NCP), we are given d-dimensional point sets P_1,...,P_n that do not necessarily contain the origin in their convex hulls. The goal is to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing local optima for the NCP problem is PLS-complete, while computing a global optimum is NP-hard.

Cite as

Wolfgang Mulzer and Yannik Stein. Computational Aspects of the Colorful Carathéodory Theorem. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 44-58, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{mulzer_et_al:LIPIcs.SOCG.2015.44,
  author =	{Mulzer, Wolfgang and Stein, Yannik},
  title =	{{Computational Aspects of the Colorful Carath\'{e}odory Theorem}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{44--58},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.44},
  URN =		{urn:nbn:de:0030-drops-51019},
  doi =		{10.4230/LIPIcs.SOCG.2015.44},
  annote =	{Keywords: colorful Carath\'{e}odory theorem, high-dimensional approximation, PLS}
}
Document
Online Train Shunting

Authors: Vianney Boeuf and Frédéric Meunier

Published in: OASIcs, Volume 42, 14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2014)


Abstract
At the occasion of ATMOS 2012, Tim Nonner and Alexander Souza defined a new train shunting problem that can roughly be described as follows. We are given a train visiting stations in a given order and cars located at some source stations. Each car has a target station. During the trip of the train, the cars are added to the train at their source stations and removed from it at their target stations. An addition or a removal of a car in the strict interior of the train incurs a cost higher than when the operation is performed at the end of the train. The problem consists in minimizing the total cost, and thus, at each source station of a car, the position the car takes in the train must be carefully decided. Among other results, Nonner and Souza showed that this problem is polynomially solvable by reducing the problem to the computation of a minimum independent set in a bipartite graph. They worked in the offline setting, i.e. the sources and the targets of all cars are known before the trip of the train starts. We study the online version of the problem, in which cars become known at their source stations. We derive a 2-competitive algorithm and prove than no better ratios are achievable. Other related questions are also addressed.

Cite as

Vianney Boeuf and Frédéric Meunier. Online Train Shunting. In 14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 42, pp. 34-45, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)


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@InProceedings{boeuf_et_al:OASIcs.ATMOS.2014.34,
  author =	{Boeuf, Vianney and Meunier, Fr\'{e}d\'{e}ric},
  title =	{{Online Train Shunting}},
  booktitle =	{14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems},
  pages =	{34--45},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-75-0},
  ISSN =	{2190-6807},
  year =	{2014},
  volume =	{42},
  editor =	{Funke, Stefan and Mihal\'{a}k, Mat\'{u}s},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2014.34},
  URN =		{urn:nbn:de:0030-drops-47512},
  doi =		{10.4230/OASIcs.ATMOS.2014.34},
  annote =	{Keywords: Bipartite graph, competitive analysis, online algorithm, train shunting problem, vertex cover}
}
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