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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…
Computing a Euclidean minimum spanning tree of a set of points is a seminal problem in computational geometry and geometric graph theory. We combine it with another classical problem in graph drawing, namely computing a monotone geometric…
Initially introduced in the framework of quantum control, the so-called "monotonic algorithms" have demonstrated excellent numerical performance when dealing with bilinear optimal control problems. This paper presents a unified formulation…
We present an $O(nrG)$ time algorithm for computing and maintaining a minimum length shortest watchman tour that sees a simple polygon under monotone visibility in direction $\theta$, while $\theta$ varies in $[0,180^{\circ})$, obtaining…
For an arbitrary tree we investigate the problems of constructing a maximum matching which minimizes or maximizes the cardinality of a maximum matching of the graph obtained from original one by its removal and present corresponding…
We perform structural and algorithmic studies of significantly generalized versions of the optimal perimeter guarding (OPG) problem. As compared with the original OPG where robots are uniform, in this paper, many mobile robots with…
We introduce and study the problem of planning a trajectory for an agent to carry out a scouting mission while avoiding being detected by an adversarial guard. This introduces an adversarial version of classical visibility-based planning…
The perfect matching complex of a simple graph $G$ is a simplicial complex having facets (maximal faces) as the perfect matchings of $G$. This article discusses the perfect matching complex of polygonal line tilings and the $\left(2 \times…
We study the problem of aggregating polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Our objective is to minimize a weighted sum of the total area and the total perimeter of the…
In this article, the issue of guarding multi-agent systems against a sequence of intruder attacks through mobile heterogeneous guards (guards with different ranges) is discussed. The article makes use of graph theoretic abstractions of such…
With the advent of autonomous robots with two- and three-dimensional scanning capabilities, classical visibility-based exploration methods from computational geometry have gained in practical importance. However, real-life laser scanning of…
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…
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.
In this extended abstract, we present a PTAS for guarding the vertices of a weakly-visible polygon $P$ from a subset of its vertices, or in other words, a PTAS for computing a minimum dominating set of the visibility graph of the vertices…
For the task of moving a set of indistinguishable agents on a connected graph with unit edge distance to an arbitrary set of goal vertices, free of collisions, we propose a fast distance optimal control algorithm that guides the agents into…
In a vertex-colored graph $G = (V, E)$, a subset $S \subseteq V$ is said to be consistent if every vertex has a nearest neighbor in $S$ with the same color. The problem of computing a minimum cardinality consistent subset of a graph is…
We study the problem of partitioning a set of $n$ objects in a metric space into $k$ clusters $V_1,\dots,V_k$. The quality of the clustering is measured by considering the vector of cluster costs and then minimizing some monotone symmetric…
In the original Art Gallery Problem (AGP), one seeks the minimum number of guards required to cover a polygon $P$. We consider the Chromatic AGP (CAGP), where the guards are colored. As long as $P$ is completely covered, the number of…
The well-known \textsc{Watchman Route} problem seeks a shortest route in a polygonal domain from which every point of the domain can be seen. In this paper, we study the cooperative variant of the problem, namely the \textsc{$k$-Watchmen…
An important task in terrain analysis is computing \emph{viewsheds}. A viewshed is the union of all the parts of the terrain that are visible from a given viewpoint or set of viewpoints. The complexity of a viewshed can vary significantly…