Polygon Containment and Translational Min-Hausdorff-Distance between Segment Sets are 3SUM-Hard
Gill Barequet, Sariel Har-Peled
TL;DR
The paper establishes that several geometric containment problems—polygon containment under translation, rotation, and rigid motion, as well as interval and segment containment on the line—are 3SUM-hard. It achieves this through a sequence of reductions from 3SUM' to interval containment (EqDist and Seg-ContPnt), then to polygon-containment variants, and finally to the problem of minimizing the Hausdorff distance between segment sets under translation. These results imply that subquadratic algorithms for these problems would yield subquadratic solutions to 3SUM, which is widely conjectured unlikely. The work maps the landscape of computational hardness for core geometric queries, using constructions like comb-shaped polygons and circular mappings to carry arithmetic structure into geometric form.
Abstract
The 3SUM problem represents a class of problems conjectured to require $Ω(n^2)$ time to solve, where $n$ is the size of the input. Given two polygons $P$ and $Q$ in the plane, we show that some variants of the decision problem, whether there exists a transformation of $P$ that makes it contained in $Q$, are 3SUM-Hard. In the first variant $P$ and $Q$ are any simple polygons and the allowed transformations are translations only; in the second and third variants both polygons are convex and we allow either rotations only or any rigid motion. We also show that finding the translation in the plane that minimizes the Hausdorff distance between two segment sets is 3SUM-Hard.