Related papers: On the Complexity of Half-Guarding Monotone Polygo…
The Art Gallery Problem (AGP) asks for placing a minimum number of stationary guards in a polygonal region P, such that all points in P are guarded. The problem is known to be NP-hard, and its inherent continuous structure (with both the…
We study the problem of planning paths for a team of robots for visually monitoring an environment. Our work is motivated by surveillance and persistent monitoring applications. We are given a set of target points in a polygonal environment…
Art Gallery Localization (AGL) is the problem of placing a set $T$ of broadcast towers in a simple polygon $P$ in order for a point to locate itself in the interior. For any point $p \in P$: for each tower $t \in T \cap V(p)$ (where $V(p)$…
We tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra.
The contiguous art gallery problem was introduced at SoCG'25 in a merged paper that combined three simultaneous results, each achieving a polynomial-time algorithm for the problem. This problem is a variant of the classical art gallery…
The Searchlight Scheduling Problem was first studied in 2D polygons, where the goal is for point guards in fixed positions to rotate searchlights to catch an evasive intruder. Here the problem is extended to 3D polyhedra, with the guards…
We devise an algorithm for surveying a dynamic orthogonal polygonal domain by placing one guard at each vertex in a subset of its vertices, i.e., whenever an orthogonal polygonal domain {\cal P'} is modified to result in another orthogonal…
We address recently proposed chromatic versions of the classic Art Gallery Problem. Assume a simple polygon $P$ is guarded by a finite set of point guards and each guard is assigned one of $t$ colors. Such a chromatic guarding is said to be…
Placing a minimum number of guards on a given watchman route in a polygonal domain is called the {\em minimum vision points problem}. We prove that finding the minimum number of vision points on a shortest watchman route in a simple polygon…
A set $G$ of points on a 1.5-dimensional terrain, also known as an $x$-monotone polygonal chain, is said to guard the terrain if any point on the terrain is 'seen' by a point in $G$. Two points on the terrain see each other if and only if…
We study the Art Gallery Problem under $k$-hop visibility in polyominoes. In this visibility model, two unit squares of a polyomino can see each other if and only if the shortest path between the respective vertices in the dual graph of the…
We provide a spectrum of results for the Universal Guard Problem, in which one is to obtain a small set of points ("guards") that are "universal" in their ability to guard any of a set of possible polygonal domains in the plane. We give…
An orthogonal polygon is called an ortho-unit polygon if its vertices have integer coordinates, and all of its edges have length one. In this paper we prove that any ortho-unit polygon with $n \geq 12$ vertices can be guarded with at most…
Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite…
We consider polynomial approximations of z-bar to better understand the torsional rigidity of polygons. Our main focus is on low degree approximations and associated extremal problems that are analogous to Polya's conjecture for torsional…
We show the following problems are in $\textsf{P}$: 1. The contiguous art gallery problem -- a variation of the art gallery problem where each guard can protect a contiguous interval along the boundary of a simple polygon. This was posed at…
Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take…
Let $P$ be an orthogonal polygon. Consider a sliding camera that travels back and forth along an orthogonal line segment $s\in P$ as its \emph{trajectory}. The camera can see a point $p\in P$ if there exists a point $q\in s$ such that $pq$…
We introduce the notion of a normal gallery, a gallery in which any configuration of guards that visually covers the walls covers the entire gallery. We show that any star gallery is normal and any gallery with at most two reflex corners is…
We solve the $r$-star covering problem in simple orthogonal polygons, also known as the point guard problem in simple orthogonal polygons with rectangular vision, in quadratic time.