Witness Set in Monotone Polygons: Exact and Approximate
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2025
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| author | Das, Udvas Dutta, Binayak Jana, Satyabrata Pal, Debabrata Roy, Sasanka |
| author_facet | Das, Udvas Dutta, Binayak Jana, Satyabrata Pal, Debabrata Roy, Sasanka |
| contents | 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 $r^{O(1/ε)} n^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10224 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Witness Set in Monotone Polygons: Exact and Approximate Das, Udvas Dutta, Binayak Jana, Satyabrata Pal, Debabrata Roy, Sasanka Computational Geometry Discrete Mathematics Combinatorics 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 $r^{O(1/ε)} n^2$. |
| title | Witness Set in Monotone Polygons: Exact and Approximate |
| topic | Computational Geometry Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2511.10224 |