Fast Approximation Algorithms for Euclidean Minimum Weight Perfect Matching

TL;DR

This work addresses the Euclidean minimum weight perfect matching problem for $n$ points in the plane (extensible to fixed dimensions) by developing near-linear-time deterministic approximations. The core method combines nearest-neighbor structure with an Even Component framework and a Node Reduction mechanism to iteratively peel off small, easily matchable parts while shrinking the remaining set. The main results are a deterministic $O(n^{0.206})$-approximation in $O(n \log n)$ time for $\mathbb{R}^2$ and a $O(n^{0.412})$-approximation in $O(n \log n)$ time for fixed dimensions, alongside a lower bound of $\Omega(n^{0.106})$ for the iterated approach. These contributions improve the longstanding $n/2$-approximation and illuminate the trade-offs between approximation quality and runtime in Euclidean matching problems.

Abstract

We study the Euclidean minimum weight perfect matching problem for $n$ points in the plane. It is known that any deterministic approximation algorithm whose approximation ratio depends only on $n$ requires at least $Ω(n \log n)$ time. We propose such an algorithm for the Euclidean minimum weight perfect matching problem with runtime $O(n\log n)$ and show that it has approximation ratio $O(n^{0.206})$. This improves the so far best known approximation ratio of $n/2$. We also develop an $O(n \log n)$ algorithm for the Euclidean minimum weight perfect matching problem in higher dimensions and show it has approximation ratio $O(n^{0.412})$ in all fixed dimensions.

Figures (4)

• Figure 1: (Top:) A $2$-well structured point set. The edges of the nearest neighbor graph $G_0$ are shown in green. The graph $G_1$ contains in addition the red edges; the graph $G_2$ contains all colored edges. The two gray edges are edges of the minimum spanning tree. (Bottom:) The rearranged point set obtained after the first step of the construction described in the proof of Lemma \ref{['lemma:structure-of-V']}.
• Figure 2: Relocation of a point set to create a 1-well structured set. a) shows the start state with the graph $G_1$ containing the red and green edges while $G_0$ contains the green edges only. In b) two odd components of $G_0$ have been relocated to the right. In c) compaction has taken place. In d) two points from an odd connected component of size 5 in $G_0$ have been relocated to the right. e) shows the final state after compaction.
• Figure 3: The recursively constructed point sets $V_i$ shown for $i=0,1,2$.
• Figure 4: A possible nearest neighbor graph for the point set $V_1$. The edges of the nearest neighbor graph are shown in red. The two points shown in red are the points that the Node-Re-duction-Algo-rithm may return as the set $W$.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4: raoSmith
• proof
• Lemma 5: raoSmith
• proof
• Lemma 6
• proof