Polygon Containment and Translational Min-Hausdorff-Distance between Segment Sets are 3SUM-Hard

2025-12-16

AI Summary

This paper investigates the computational complexity of two classical computational geometry problems: (1) polygon containment under transformations—determining whether polygon $P$ can be placed inside polygon $Q$ via translation, rotation, or rigid motion; and (2) translation optimization for minimizing the Hausdorff distance between two sets of line segments. We establish the first systematic reduction framework linking both problems to the 3SUM conjecture. Under multiple transformation models—including translation-only, rotation-only, and full rigid motion—we rigorously prove 3SUM-hardness, thereby establishing an $Omega(n^2)$ conditional lower bound and ruling out subquadratic algorithms. Our key innovation lies in a geometric reduction technique that jointly leverages piecewise-linear structural properties and rigid-motion parameterization. This unified approach reveals the intrinsic quadratic complexity shared by both polygon containment and Hausdorff distance optimization, providing a foundational explanation for their persistent algorithmic hardness.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Determining polygon containment under translation is 3SUM-hard.
• Finding rotational containment for convex polygons is 3SUM-hard.
• Minimizing Hausdorff distance between segment sets is 3SUM-hard.

Innovation

• Proves polygon containment variants are 3SUM-hard
• Shows translational Hausdorff distance is 3SUM-hard
• Establishes hardness for translations, rotations, rigid motions