Geometric Bipartite Matching is in NC

TL;DR

This paper investigates the parallel computational complexity of the Euclidean minimum-weight perfect matching problem, establishing its inclusion in NC when allowing a small additive error, thus advancing understanding of its parallel solvability.

Contribution

The paper proves that Euclidean minimum-weight perfect matching with a small additive error is in NC, addressing a long-standing open problem in parallel complexity.

Findings

Minimum-weight perfect matching requires linear bits of approximation to distinguish from others.

EWPM with up to 1/poly(n) additive error is in NC.

Advances understanding of parallel complexity for geometric matching problems.

Abstract

In this work, we study the parallel complexity of the Euclidean minimum-weight perfect matching (EWPM) problem. Here our graph is the complete bipartite graph $G$ on two sets of points $A$ and $B$ in $\mathbb{R}^2$ and the weight of each edge is the Euclidean distance between the corresponding points. The weighted perfect matching problem on general bipartite graphs is known to be in RNC [Mulmuley, Vazirani, and Vazirani, 1987], and Quasi-NC [Fenner, Gurjar, and Thierauf, 2016]. Both of these results work only when the weights are of $O(\log n)$ bits. It is a long-standing open question to show the problem to be in NC. First, we show that for EWPM, a linear number of bits of approximation is required to distinguish between the minimum-weight perfect matching and other perfect matchings. Next, we show that the EWPM problem that allows up to $\frac{1}{poly(n)}$ additive error, is in NC.