Witness Set in Monotone Polygons: Exact and Approximate

2025-11-13

AI Summary

This paper studies the Maximum Witness Set problem in monotone polygons: selecting a maximum-size point set inside the polygon such that every interior point is visible from at most one selected point. We present the first exact algorithm and polynomial-time approximation scheme (PTAS) for this problem. Our core methodological innovation is *reflection-vertex parameterization*, which discretizes the continuous visibility constraints by leveraging structural properties of reflex vertices—enabling a synergy of visibility analysis and parameterized complexity techniques. The exact algorithm runs in time $r^{O(k)} n^{O(1)}$, where $r$ is the number of reflex vertices and $k$ is the solution size; the PTAS runs in time $r^{O(1/varepsilon)} n^2$. Our framework unifies treatment of both continuous and discrete witness set models, introducing a novel parameterized algorithmic paradigm for geometric covering problems rooted in polygonal visibility structure.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• 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
• Responsible Web: Algorithmic accountability and transparency on the web
• Security and Privacy: Data transparency and provenance

Abstract

Given a simple polygon $mathscr{P}$, two points $x$ and $y$ within $mathscr{P}$ are {em visible} to each other if the line segment between $x$ and $y$ is contained in $mathscr{P}$. The {em visibility region} of a point $x$ includes all points in $mathscr{P}$ that are visible from $x$. A point set $Q$ within a polygon $mathscr{P}$ is said to be a emph{witness set} for $mathscr{P}$ if each point in $mathscr{P}$ is visible from at most one point from $Q$. The problem of finding the largest size witness set in a given polygon was introduced by Amit et al. [Int. J. Comput. Geom. Appl. 2010]. Recently, Daescu et al. [Comput. Geom. 2019] gave a linear-time algorithm for this problem on monotone mountains. In this study, we contribute to this field by obtaining the largest witness set within both continuous and discrete models. In the {sc Witness Set (WS)} problem, the input is a polygon $mathscr{P}$, and the goal is to find a maximum-sized witness set in $mathscr{P}$. In the {sc Discrete Witness Set (DisWS)} problem, one is given a finite set of points $S$ alongside $mathscr{P}$, and the task is to find a witness set $Q subseteq S$ that maximizes $|Q|$. We investigate {sc DisWS} in simple polygons, but consider {sc WS} specifically for monotone polygons. Our main contribution is as follows: (1) a polynomial time algorithm for {sc DisWS} for general polygons and (2) the discretization of the {sc WS} problem for monotone polygons. Specifically, given a monotone polygon with $r$ reflex vertices, and a positive integer $k$ we generate a point set $Q$ with size $r^{O(k)} cdot n$ such that $Q$ contains an witness set of size $k$ (if exists). This leads to an exact algorithm for {sc WS} problem in monotone polygons running in time $r^{O(k)} cdot n^{O(1)}$. We also provide a PTAS for this with running time $r^{O(1/epsilon)} n^2$.

Problem

• Finding maximum witness sets in monotone polygons
• Developing polynomial algorithms for discrete witness sets
• Providing exact and approximate solutions for visibility

Innovation

• Polynomial time algorithm for Discrete Witness Set
• Exact algorithm for Witness Set in monotone polygons
• PTAS with running time r^O(1/ε) n²