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DOI: 10.4230/LIPIcs.SoCG.2019.15
Preconditioning for the Geometric Transportation Problem

Authors: Andrey Boris Khesin, Aleksandar Nikolov, and Dmitry Paramonov

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
In the geometric transportation problem, we are given a collection of points P in d-dimensional Euclidean space, and each point is given a supply of mu(p) units of mass, where mu(p) could be a positive or a negative integer, and the total sum of the supplies is 0. The goal is to find a flow (called a transportation map) that transports mu(p) units from any point p with mu(p) > 0, and transports -mu(p) units into any point p with mu(p) < 0. Moreover, the flow should minimize the total distance traveled by the transported mass. The optimal value is known as the transportation cost, or the Earth Mover’s Distance (from the points with positive supply to those with negative supply). This problem has been widely studied in many fields of computer science: from theoretical work in computational geometry, to applications in computer vision, graphics, and machine learning. In this work we study approximation algorithms for the geometric transportation problem. We give an algorithm which, for any fixed dimension d, finds a (1+epsilon)-approximate transportation map in time nearly-linear in n, and polynomial in epsilon^{-1} and in the logarithm of the total supply. This is the first approximation scheme for the problem whose running time depends on n as n * polylog(n). Our techniques combine the generalized preconditioning framework of Sherman, which is grounded in continuous optimization, with simple geometric arguments to first reduce the problem to a minimum cost flow problem on a sparse graph, and then to design a good preconditioner for this latter problem.
Cite as

Andrey Boris Khesin, Aleksandar Nikolov, and Dmitry Paramonov. Preconditioning for the Geometric Transportation Problem. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{khesin_et_al:LIPIcs.SoCG.2019.15,
  author =	{Khesin, Andrey Boris and Nikolov, Aleksandar and Paramonov, Dmitry},
  title =	{{Preconditioning for the Geometric Transportation Problem}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{15:1--15:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.15},
  URN =		{urn:nbn:de:0030-drops-104190},
  doi =		{10.4230/LIPIcs.SoCG.2019.15},
  annote =	{Keywords: Earth Mover Distance, Transportation Problem, Minimum Cost Flow}
}
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