Euclidean Maximum Matchings in the Plane---Local to Global

TL;DR

We study how well $k$-local maximum matchings approximate the global maximum in the Euclidean plane, focusing on $k=2$ and $k=3$. The authors develop a geometric framework based on diametral disks, Helly-type intersection results, and distance-geometry lemmas to connect local optimality to global optimality, yielding improved lower bounds $\mu_2 \ge \sqrt{3/7}$ and $\mu_3 \ge \sqrt{3}/2$ (with an intermediate $1/\sqrt{2}$ bound for $3$-local). They also provide explicit upper-bound examples $\mu_2<0.93$ and $\mu_3<0.98$, and prove that any pairwise-crossing matching is globally maximum and unique via a reduction to edge-disjoint paths in planar graphs (Okamura–Seymour). These results advance understanding of local-to-global behavior in geometric matching and have implications for local-search approximation strategies and potential efficient algorithms.

Abstract

Let $M$ be a perfect matching on a set of points in the plane where every edge is a line segment between two points. We say that $M$ is globally maximum if it is a maximum-length matching on all points. We say that $M$ is $k$-local maximum if for any subset $M'=\{a_1b_1,\dots,a_kb_k\}$ of $k$ edges of $M$ it holds that $M'$ is a maximum-length matching on points $\{a_1,b_1,\dots,a_k,b_k\}$. We show that local maximum matchings are good approximations of global ones. Let $μ_k$ be the infimum ratio of the length of any $k$-local maximum matching to the length of any global maximum matching, over all finite point sets in the Euclidean plane. It is known that $μ_k\geqslant \frac{k-1}{k}$ for any $k\geqslant 2$. We show the following improved bounds for $k\in\{2,3\}$: $\sqrt{3/7}\leqslantμ_2< 0.93 $ and $\sqrt{3}/2\leqslantμ_3< 0.98$. We also show that every pairwise crossing matching is unique and it is globally maximum. Towards our proof of the lower bound for $μ_2$ we show the following result which is of independent interest: If we increase the radii of pairwise intersecting disks by factor $2/\sqrt{3}$, then the resulting disks have a common intersection.

Figures (8)

• Figure 1: Illustration of upper bounds for (a) 2-local matchings and (b) 3-local matchings.
• Figure 2: Red edges belong to $M$, black edges belong to $S$, and blue edge belongs to $M^*$.
• Figure 3: Illustration of the proof of Lemma \ref{['endpoint-lemma']}.
• Figure 4: Illustration of the proof of Lemma \ref{['diameter-lemma']}.
• Figure 5: Illustration of the proof of Theorem \ref{['stretch-lemma']}

Theorems & Definitions (20)

• Theorem 1
• proof
• Theorem 2: Bereg et al. Bereg2019
• Theorem 3: Helly's theorem in $\mathcal{R}^2$
• Theorem 4
• proof
• Theorem 5
• Lemma 1
• proof
• Lemma 2