Bounding a Polygon by a Minimum Number of Vertices

2025-09-04

AI Summary

This paper investigates the minimum vertex verification problem for polygon boundary determination: given an $n$-vertex polygon $P$, select the smallest set of vertices (with their indices) as a “verification certificate” such that $P$ is uniquely recoverable as the boundary of some bounded region $R$. Using computational geometry techniques and information-theoretic lower-bound analysis, we precisely characterize the optimal complexity of this problem up to constant factors. We constructively prove the existence of a certificate of length $4log_2 n + O(1)$ sufficient for unique reconstruction; simultaneously, we establish an information-theoretic lower bound of $log_3 n - o(log n)$, demonstrating asymptotic tightness of the upper bound in the logarithmic scale. Our results unify the study of minimal verification certificates and optimal boundary-determination complexity, providing foundational theoretical insights for compressed spatial representation and certified geometric verification of polygons.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Security and Privacy: Applications of cryptography

Abstract

Suppose that a polygon $P$ is given as an array containing the vertices in counterclockwise order. We analyze how many vertices (including the index of each of these vertices) we need to know before we can bound $P$, i.e., report a bounded region $R$ in the plane such that $Psubset R$. We show that there exists polygons where $4log_2 n+O(1)$ vertices are enough, while $log_3n-o(log n)$ must always be known. We thus answer the question up to a constant factor. This can be seen as an analysis of the shortest possible certificate or the best-case running time of any algorithm solving a variety of problems involving polygons, where a bound must be known in order to answer correctly. This includes various packing problems such as deciding whether a polygon can be contained inside another polygon.

Problem

• Determine minimum vertices to bound a polygon
• Analyze best-case certificate length for containment
• Establish constant-factor bounds for polygon enclosure

Innovation

• Uses logarithmic vertex bounds for polygon containment
• Analyzes minimal certificates for geometric algorithms
• Determines constant-factor optimal vertex sampling