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Fast Algorithms for Continuous Optimal Transport Between Histograms

Authors Pankaj K. Agarwal , Sharath Raghvendra, Keegan Yao

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Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Computational geometry
Keywords
  • Optimal transport
  • min-cost flow
  • minimum-weight matching
  • quadtrees
  • approximation algorithms
  • dynamic optimal transport algorithm

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Abstract

We give the first (relative) (1+ε)-approximation algorithm for the continuous optimal transport (OT) problem between two axis-aligned histograms in the plane, each of which is represented as a piecewise-constant function over a rectangular subdivision. Our algorithm runs in nearly quadratic time in the complexity of the histograms in the worst case. In contrast to the discrete and semi-discrete optimal transport problems, which always admit linear-size OT plans, we demonstrate that there is a quadratic lower bound on the complexity of even a (1+ε)-approximate OT plan in the continuous setting in the worst case. This suggests that the runtime of our approximation algorithm nearly matches the worst-case lower-bound complexity of an explicit (1+ε)-approximate OT plan between two histograms. We additionally provide near-linear time algorithms with either weaker approximation guarantees or restrictions on the input histograms.

Cite As Get BibTex

Pankaj K. Agarwal, Sharath Raghvendra, and Keegan Yao. Fast Algorithms for Continuous Optimal Transport Between Histograms. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 123:1-123:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.123

Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Sharath Raghvendra
  • Department of Computer Science, North Carolina State University, Raleigh, NC, USA
Keegan Yao
  • Department of Computer Science, Duke University, Durham, NC, USA

Funding

  • Agarwal, Pankaj K.: NSF grants CCF-22-23870 and IIS-24-02823, and US-Israel Binational Science Foundation Grant 2022131.
  • Raghvendra, Sharath: NSF grant CCF-2514753.
  • Yao, Keegan: NSF grants CCF-22-23870 and IIS-24-02823, and US-Israel Binational Science Foundation Grant 2022131.

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