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An Improved ε-Approximation Algorithm for Geometric Bipartite Matching

Authors Pankaj K. Agarwal, Sharath Raghvendra, Pouyan Shirzadian, Rachita Sowle



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Author Details

Pankaj K. Agarwal
  • Duke University, Durham, NC, USA
Sharath Raghvendra
  • Virginia Tech, Blacksburg, VA, USA
Pouyan Shirzadian
  • Virginia Tech, Blacksburg, VA, USA
Rachita Sowle
  • Virginia Tech, Blacksburg, VA, USA

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Pankaj K. Agarwal, Sharath Raghvendra, Pouyan Shirzadian, and Rachita Sowle. An Improved ε-Approximation Algorithm for Geometric Bipartite Matching. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 6:1-6:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SWAT.2022.6

Abstract

For two point sets A, B ⊂ ℝ^d, with |A| = |B| = n and d > 1 a constant, and for a parameter ε > 0, we present a randomized algorithm that, with probability at least 1/2, computes in O(n(ε^{-O(d³)}log log n + ε^{-O(d)}log⁴ nlog⁵log n)) time, an ε-approximate minimum-cost perfect matching under any L_p-metric. All previous algorithms take n(ε^{-1}log n)^{Ω(d)} time. We use a randomly-shifted tree, with a polynomial branching factor and O(log log n) height, to define a tree-based distance function that ε-approximates the L_p metric as well as to compute the matching hierarchically. Then, we apply the primal-dual framework on a compressed representation of the residual graph to obtain an efficient implementation of the Hungarian-search and augment operations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Euclidean bipartite matching
  • approximation algorithms
  • primal dual method

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