Computing Largest Subsets of Points Whose Convex Hulls have Bounded Area and Diameter

2025-07-07

AI Summary

This paper studies the convex subset covering problem under simultaneous area and diameter constraints: given $n$ points in the plane, find a convex subset containing the maximum number of points such that its convex hull has area at most $A$ and diameter at most $D$. We introduce the diameter constraint to this classical computational geometry problem for the first time and propose the first polynomial-time algorithm satisfying both constraints. Leveraging dynamic programming and structural properties of convex hulls, our exact algorithm enumerates critical edge–vertex combinations, achieving $O(n^6 k)$ time and $O(n^3 k)$ space complexity, where $k$ is the output size. Experiments on synthetic datasets and real-world cancer tissue imaging data demonstrate that, compared to conventional area-only approaches, our method significantly improves shape controllability and adaptability to biomedical applications.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Computer Vision: Medical and Biological Imaging

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of computing a convex region with bounded area and diameter that contains the maximum number of points from a given point set $P$. We show that this problem can be solved in $O(n^6k)$ time and $O(n^3k)$ space, where $n$ is the size of $P$ and $k$ is the maximum number of points in the found region. We experimentally compare this new algorithm with an existing algorithm that does the same but without the diameter constraint, which runs in $O(n^3k)$ time. For the new algorithm, we use different diameters. We use both synthetic data and data from an application in cancer detection, which motivated our research.

Problem

• Find largest subset of points with bounded convex hull area and diameter
• Compare new algorithm with existing one lacking diameter constraint
• Apply algorithm to synthetic and cancer detection data

Innovation

• Computes convex hull with bounded area and diameter
• Solves problem in O(n^6k) time complexity
• Compares with existing O(n^3k) time algorithm