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

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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.

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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

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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