# Efficient Algorithms for Geometric Partial Matching

## TL;DR

This paper introduces efficient algorithms for the geometric partial matching problem in the plane, providing exact and approximate solutions with improved runtime using primal-dual flow augmentation and dynamic data structures.

## Contribution

The paper presents novel algorithms for geometric partial matching that achieve faster runtimes for exact and approximate solutions, applicable to any $L_p$-norm matching objectives.

## Key findings

- Exact algorithm runs in $O((n + k^2) ext{polylog} n)$ time.
- Approximate $(1 + ext{varepsilon})$-algorithm runs in $O((n + k extsqrt{k}) ext{polylog} n imes ext{log} extvarepsilon^{-1})$ time.
- Additional exact algorithm for planar transportation problem with $O( extminig
^2, r n^{3/2}ig) ext{polylog} n$ runtime.

## Abstract

Let $A$ and $B$ be two point sets in the plane of sizes $r$ and $n$ respectively (assume $r \leq n$), and let $k$ be a parameter. A matching between $A$ and $B$ is a family of pairs in $A \times B$ 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 $\|{\cdot}\|_p$ 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 $O((n + k^2) {\mathop{\mathrm{polylog}}} n)$ time, and a $(1 + \varepsilon)$-approximation algorithm that runs in $O((n + k\sqrt{k}) {\mathop{\mathrm{polylog}}} n \cdot \log\varepsilon^{-1})$ 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 $O(\min\{n^2, rn^{3/2}\} {\mathop{\mathrm{polylog}}} n)$ time.