A constant-factor approximation of the Gromov-Hausdorff distance in the plane

2026-06-15

AI Summary

This work presents the first polynomial-time constant-factor approximation algorithm for the Gromov–Hausdorff distance in the Euclidean plane, resolving a long-standing open problem in two dimensions. The approach introduces the bijective Gromov–Hausdorff distance and combines rigid alignment, multiscale tree-based clustering matching, and reflection sign determination within a novel “fat-or-collinear” dichotomy framework. Key technical contributions include a relaxation of correspondences, an additive distortion approximation scheme, and unified theoretical guarantees that bridge bijective and general correspondence settings. The analysis further establishes the necessity of several structural barriers—such as dimensionality reduction, multiplicity gaps, and reflection obstructions—for any lower-bound arguments, thereby shedding light on fundamental limitations inherent to the problem.

Abstract

We give the first polynomial-time constant-factor approximation of the Gromov--Hausdorff distance $d_{GH}$ between finite point sets in the Euclidean plane; in fixed Euclidean dimension such an approximation was previously known only on the line (Majhi, Vitter, and Wenk, 2024). Its engine is the bijective (bottleneck) Gromov--Hausdorff distance $d_{GH}^{bij}$: for two equal-size sets the least additive distortion $\max_{i,j}|d_X(i,j) - d_Y(σi, σj)|$ of a bijection $σ$ equals $2\,d_{GH}^{bij}$, which we likewise approximate within an absolute constant. Approximating additive distortion goes back to Hall and Papadimitriou (2005), who gave a $2$-approximation on the line and observed approximation within $3$ to be NP-hard in dimension three; the planar case they left open is the one we settle. A fat-or-collinear dichotomy drives both bounds: a fat set is aligned by a single rigid motion, while a near-collinear set is split into clusters matched along their dendrogram in one flat, scale-free pass, with relative orientations and per-node reflection signs -- at every scale of the dendrogram -- recovered by global cuts. Relaxing bijections to correspondences yields $d_{GH}$ itself, which reduces to a lone within-cluster-multiplicity kernel -- the pairs an optimal correspondence collapses -- that the same theory closes. Matching lower bounds -- a dimension drop, a multiplicity gap, and a reflection barrier acting at every scale -- show each ingredient is necessary.

Problem

• Gromov-Hausdorff distance
• constant-factor approximation
• finite point sets
• Euclidean plane
• additive distortion

Innovation

• Gromov-Hausdorff distance
• constant-factor approximation
• planar point sets
• bijective distortion
• dendrogram-based matching