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no code implementations • 14 Mar 2022 • Peyman Afshani, Mark De Berg, Kevin Buchin, Jie Gao, Maarten Loffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, Hao-Tsung Yang

For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a $(2(1-1/k)+\varepsilon)$-approximation of the optimal unrestricted solution.

no code implementations • NeurIPS 2021 • Rishi Sonthalia, Gregory Van Buskirk, Benjamin Raichel, Anna C. Gilbert

While $D_l$ is not metric, when given as input to cMDS instead of $D$, it empirically results in solutions whose distance to $D$ does not increase when we increase the dimension and the classification accuracy degrades less than the cMDS solution.

no code implementations • 5 May 2020 • Peyman Afshani, Mark De Berg, Kevin Buchin, Jie Gao, Maarten Loffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, Hao-Tsung Yang

The problem is NP-hard, as it has the traveling salesman problem as a special case (when $k=1$ and all sites have the same weight).

no code implementations • 24 Apr 2020 • Kevin Buchin, Chenglin Fan, Maarten Löffler, Aleksandr Popov, Benjamin Raichel, Marcel Roeloffzen

We prove that both the upper and lower bound problems are NP-hard for the continuous Fr\'echet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Fr\'echet distance.

Computational Geometry

no code implementations • NeurIPS 2017 • Greg Van Buskirk, Benjamin Raichel, Nicholas Ruozzi

Equivalently, given the matrix X, consider the problem of finding a small subset, S, of the columns of X such that the conic hull of S \eps-approximates the conic hull of the columns of X, i. e., the distance of every column of X to the conic hull of the columns of S should be at most an \eps-fraction of the angular diameter of X.

no code implementations • 9 Jul 2015 • Avrim Blum, Sariel Har-Peled, Benjamin Raichel

]{#1\left({#2}\right)} \newcommand{\npoints}{n} \newcommand{\ballD}{\mathsf{b}} \newcommand{\dataset}{{P}} $ For a set $\dataset$ of $\npoints$ points in the unit ball $\ballD \subseteq \Re^d$, consider the problem of finding a small subset $\algset \subseteq \dataset$ such that its convex-hull $\eps$-approximates the convex-hull of the original set.

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