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Efficient Algorithms for Geometric Partial Matching

Authors Pankaj K. Agarwal, Hsien-Chih Chang, Allen Xiao

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • partial matching
  • transportation
  • optimal transport
  • minimum-cost flow
  • bichromatic closest pair

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Abstract

Let A and B be two point sets in the plane of sizes r and n respectively (assume r <= n), and let k be a parameter. A matching between A and B is a family of pairs in A x B so that any point of A cup B appears in at most one pair. Given two positive integers p and q, we define the cost of matching M to be c(M) = sum_{(a, b) in M}||a-b||_p^q where ||*||_p is the L_p-norm. The geometric partial matching problem asks to find the minimum-cost size-k matching between A and B.
We present efficient algorithms for geometric partial matching problem that work for any powers of L_p-norm matching objective: An exact algorithm that runs in O((n + k^2)polylog n) time, and a (1 + epsilon)-approximation algorithm that runs in O((n + k sqrt{k})polylog n * log epsilon^{-1}) time. Both algorithms are based on the primal-dual flow augmentation scheme; the main improvements involve using dynamic data structures to achieve efficient flow augmentations. With similar techniques, we give an exact algorithm for the planar transportation problem running in O(min{n^2, rn^{3/2}}polylog n) time.

Cite As Get BibTex

Pankaj K. Agarwal, Hsien-Chih Chang, and Allen Xiao. Efficient Algorithms for Geometric Partial Matching. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.6

Author Details

Pankaj K. Agarwal
  • Duke University, Durham, USA
Hsien-Chih Chang
  • Duke University, Durham, USA
Allen Xiao
  • Duke University, Durham, USA

Funding

Work on this paper was supported by NSF under grants CCF-15-13816, CCF-15-46392, and IIS-14-08846, by an ARO grant W911NF-15-1-0408, and by BSF Grant 2012/229 from the U.S.-Israel Binational Science Foundation.

Acknowledgements

We thank Haim Kaplan for discussion and suggestion to use Goldberg et al. [Andrew V. Goldberg et al., 2017] algorithm. We thank Debmalya Panigrahi for helpful discussions.

References

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