Computing the Gromov-Hausdorff Distance for Metric Trees

The Gromov-Hausdorff distance is a natural way to measure distance between two metric spaces. We give the first proof of hardness and first non-trivial approximation algorithm for computing the Gromov-Hausdorff distance for geodesic metrics in trees. Specifically, we prove it is NP-hard to approximate the Gromov-Hausdorff distance better than a factor of 3. We complement this result by providing a polynomial time O(min{n, √ rn})-approximation algorithm where r is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an O( √ n)-approximation algorithm. ∗Work on this paper by P. K. Agarwal, K. Fox and A. Nath was supported by NSF under grants CCF-09-40671, CCF10-12254, CCF-11-61359, and IIS-14-08846, and by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. A. Sidiropoulos was supported by NSF under grants CAREER-1453472 and CCF-1423230. Y. Wang was supported by NSF under grant CCF–1319406.