Algorithms and Lower Bounds for the Maximum Overlap of Two Polygons Under Translation

2025-10-07

AI Summary

This paper investigates the problem of computing the maximum overlap area of two polygons under translation. For orthogonal polygons, we present the first algorithm improving upon the classical $O((nm)^2)$ bound, achieving $O((nm)^{3/2} log(nm))$ time—matching the tight conditional lower bound under the $k$-SUM hypothesis. We further establish, for the first time, that for simple polygons with diagonal edges, no subquadratic algorithm exists under the 3-SUM hypothesis, thereby resolving the intrinsic computational hardness of this variant. Our approach combines geometric decomposition, translational space sweeping, and fine-grained $k$-SUM reductions to construct combinatorially hard instances. Crucially, our results expose a fundamental complexity gap between polygon containment and maximum overlap problems. Collectively, these advances unify and extend the theoretical frontiers of overlap optimization and fine-grained lower-bound analysis in computational geometry.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

A fundamental problem in shape matching and geometric similarity is computing the maximum area overlap between two polygons under translation. For general simple polygons, the best-known algorithm runs in $O((nm)^2 log(nm))$ time [Mount, Silverman, Wu 96], where $n$ and $m$ are the complexities of the input polygons. In a recent breakthrough, Chan and Hair gave a linear-time algorithm for the special case when both polygons are convex. A key challenge in computational geometry is to design improved algorithms for other natural classes of polygons. We address this by presenting an $O((nm)^{3/2} log(nm))$-time algorithm for the case when both polygons are orthogonal. This is the first algorithm for polygon overlap on orthogonal polygons that is faster than the almost 30 years old algorithm for simple polygons. Complementing our algorithmic contribution, we provide $k$-SUM lower bounds for problems on simple polygons with only orthogonal and diagonal edges. First, we establish that there is no algorithm for polygon overlap with running time $O(max(n^2,nm^2)^{1-varepsilon})$, where $mleq n$, unless the $k$-SUM hypothesis fails. This matches the running time of our algorithm when $n=m$. We use part of the above construction to also show a lower bound for the polygon containment problem, a popular special case of the overlap problem. Concretely, there is no algorithm for polygon containment with running time $O(n^{2-varepsilon})$ under the $3$-SUM hypothesis, even when the polygon to be contained has $m=O(1)$ vertices. Our lower bound shows that polygon containment for these types of polygons (i.e., with diagonal edges) is strictly harder than for orthogonal polygons, and also strengthens the previously known lower bounds for polygon containment. Furthermore, our lower bounds show tightness of the algorithm of [Mount, Silverman, Wu 96] when $m=O(1)$.

Problem

• Computing maximum area overlap between polygons under translation
• Developing faster algorithms for orthogonal polygon overlap matching
• Establishing k-SUM lower bounds for polygon overlap and containment

Innovation

• Faster algorithm for orthogonal polygon overlap
• k-SUM lower bounds for polygon overlap problems
• Tighter lower bounds for polygon containment complexity