The Gromov--Hausdorff distance measures the difference in shape between compact metric spaces and poses a notoriously difficult problem in combinatorial optimization. We introduce its quadratic relaxation over a convex polytope whose solutions provably deliver the Gromov--Hausdorff distance. The optimality guarantee is enabled by the fact that the search space of our approach is not constrained to a generalization of bijections, unlike in other relaxations such as the Gromov--Wasserstein distance. We suggest the Frank--Wolfe algorithm for solving the relaxation in $O(n^3)$ time per iteration, and numerically demonstrate its performance on metric spaces of hundreds of points. In particular, we use it to obtain a new bound of the Gromov--Hausdorff distance between the unit circle and the unit hemisphere equipped with Euclidean metric. Our approach is implemented as a Python package dGH.

翻译：Gromov–Hausdorff距离衡量紧致度量空间之间的形状差异，是组合优化中公认的难题。我们提出了其在一个凸多面体上的二次松弛，该松弛的解能够可靠地给出Gromov–Hausdorff距离。与Gromov–Wasserstein距离等其他松弛方法不同，我们的搜索空间不局限于双射的推广，从而保证了最优性。我们建议使用Frank–Wolfe算法求解该松弛，每次迭代的时间复杂度为$O(n^3)$，并在包含数百个点的度量空间上数值验证了其性能。特别地，我们利用该方法获得了单位圆与配备欧几里得度量的单位半球之间Gromov–Hausdorff距离的一个新上界。我们的方法已实现为Python包dGH。