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Certified Approximation Algorithms for the Fermat Point and n-Ellipses

Authors Kolja Junginger, Ioannis Mantas , Evanthia Papadopoulou , Martin Suderland , Chee Yap

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Author Details

Kolja Junginger
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Ioannis Mantas
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Evanthia Papadopoulou
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Martin Suderland
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Chee Yap
  • Courant Institute, New York University, NY, USA

Funding

The work of C. Yap was supported by an USI Visiting Professor Fellowship (Fall 2018); additional support came from NSF Grants # CCF-1564132 and # CCF-2008768. The other authors were partially supported by the SNF project 200021E_154387.

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Kolja Junginger, Ioannis Mantas, Evanthia Papadopoulou, Martin Suderland, and Chee Yap. Certified Approximation Algorithms for the Fermat Point and n-Ellipses. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 54:1-54:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ESA.2021.54

Abstract

Given a set A of n points in ℝ^d with weight function w: A→ℝ_{> 0}, the Fermat distance function is φ(x): = ∑_{a∈A}w(a)‖x-a‖. A classic problem in facility location dating back to 1643, is to find the Fermat point x*, the point that minimizes the function φ. We consider the problem of computing a point x̃* that is an ε-approximation of x* in the sense that ‖x̃*-x*‖<ε. The algorithmic literature has so far used a different notion based on ε-approximation of the value φ(x*). We devise a certified subdivision algorithm for computing x̃*, enhanced by Newton operator techniques. We also revisit the classic Weiszfeld-Kuhn iteration scheme for x*, turning it into an ε-approximate Fermat point algorithm. Our second problem is the certified construction of ε-isotopic approximations of n-ellipses. These are the level sets φ^{-1}(r) for r > φ(x*) and d = 2. Finally, all our planar (d = 2) algorithms are implemented in order to experimentally evaluate them, using both synthetic as well as real world datasets. These experiments show the practicality of our techniques.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Fermat point
  • n-ellipse
  • subdivision
  • approximation
  • certified
  • algorithms

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