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.

翻译：我们给出了欧几里得平面中有穷点集间Gromov-Hausdorff距离$d_{GH}$的首个多项式时间常数因子近似；在固定欧几里得维度下，此前仅在直线上存在此类近似（Majhi, Vitter, and Wenk, 2024）。其核心是双射（瓶颈）Gromov-Hausdorff距离$d_{GH}^{bij}$：对于两个等势集，双射$\sigma$的最小加性畸变$\max_{i,j}|d_X(i,j) - d_Y(\sigma i, \sigma j)|$等于$2\,d_{GH}^{bij}$，我们同样对该距离在绝对常数范围内实现近似。加性畸变近似可追溯至Hall和Papadimitriou（2005），他们在直线上给出了$2$-近似，并指出在三维空间中常数$3$的近似是NP难的；他们遗留的平面情形正是本文解决的问题。胖集-共线二分法驱动两类下界：胖集可通过单一刚体运动对齐，而近共线集则沿其树状图在一次无标度平坦扫描中被分割为簇，并在树状图的每个尺度上通过全局割恢复相对朝向与每节点反射符号。将双射松弛为对应关系可得到$d_{GH}$本身，该问题约化为仅含簇内多重性核——最优对应折叠的配对——而同一理论封闭了该问题。匹配下界（维度下降、多重性间隙及作用于每个尺度的反射障碍）表明每个成分都是必要的。