Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs

AbstractIn this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs.We transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that: (1) ⌊n+13⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊2n+15⌋ edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: (1) a diagonal 2-dominating set of size ⌊n+13⌋ in linear time and space, (2) an edge 2-dominating set of size ⌊2n+15⌋ in O(n2) time and O(n) space, and (3) an edge 2-dominating set of size ⌊3n7⌋ in O(n) time and space.Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: (1) a mobile guard set of size ⌊n+13⌋ in O(nlogn) time, (2) an edge guard set of size ⌊2n+15⌋ in O(n2) time, and (3) an edge guard set of size ⌊3n7⌋ in O(nlogn) time. All space requirements are linear. Finally, we show that ⌊n3⌋ mobile or ⌈n3⌉ edge guards are sometimes necessary.When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ⌈n+14⌉ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ⌈n+14⌉, can be computed in O(n) time and space

Connectivity Constraints in Network Analysis

This dissertation establishes mathematical foundations of connectivity requirements arising in both abstract and geometric network analysis. Connectivity constraints are ubiquitous in network design and network analysis. Aside from the obvious applications in communication and transportation networks, they have also appeared in forest planning, political distracting, activity detection in video sequences and protein-protein interaction networks. Theoretically, connectivity constraints can be analyzed via polyhedral methods, in which we investigate the structure of (vertex)-connected subgraph polytope (CSP). One focus of this dissertation is on performing an extensive study of facets of CSP. We present the first systematic study of non-trivial facets of CSP. One advantage to study facets is that a facet-defining inequality is always among the tightest valid inequalities, so applying facet-defining inequalities when imposing connectivity constraints can guarantee good performance of the algorithm. We adopt lifting techniques to provide a framework to generate a wide class of facet-defining inequalities of CSP. We also derive the necessary and sufficient conditions when a vertex separator inequality, which plays a critical role in connectivity constraints, induces a facet of CSP. Another advantage to study facets is that CSP is uniquely determined by its facets, so full understanding of CSP's facets indicates full understanding of CSP itself. We are able to derive a full description of CSP for a wide class of graphs, including forest and several types of dense graphs, such as graphs with small independence number, s-plex with small s and s-defective cliques with small s. Furthermore, we investigate the relationship between lifting techniques, maximum weight connected subgraph problem and node-weight Steiner tree problem and study the computational complexity of generation of facet-defining inequalities. Another focus of this dissertation is to study connectivity in geometric network analysis. In geometric applications like wireless networks and communication networks, the concept of connectivity can be defined in various ways. In one case, connectivity is imposed by distance, which can be modeled by unit disk graphs (UDG). We create a polytime algorithm to identify large 2-clique in UDG; in another case when connectivity is based on visibility, we provide a generalization of the two-guard problem

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

Introduction: Ways of Machine Seeing

How do machines, and, in particular, computational technologies, change the way we see the world? This special issue brings together researchers from a wide range of disciplines to explore the entanglement of machines and their ways of seeing from new critical perspectives. This 'editorial' is for a special issue of AI & Society, which includes contributions from: María Jesús Schultz Abarca, Peter Bell, Tobias Blanke, Benjamin Bratton, Claudio Celis Bueno, Kate Crawford, Iain Emsley, Abelardo Gil-Fournier, Daniel Chávez Heras, Vladan Joler, Nicolas Malevé, Lev Manovich, Nicholas Mirzoeff, Perle Møhl, Bruno Moreschi, Fabian Offert, Trevor Paglan, Jussi Parikka, Luciana Parisi, Matteo Pasquinelli, Gabriel Pereira, Carloalberto Treccani, Rebecca Uliasz, and Manuel van der Veen

Unmet goals of tracking: within-track heterogeneity of students' expectations for

Educational systems are often characterized by some form(s) of ability grouping, like tracking. Although substantial variation in the implementation of these practices exists, it is always the aim to improve teaching efficiency by creating homogeneous groups of students in terms of capabilities and performances as well as expected pathways. If students’ expected pathways (university, graduate school, or working) are in line with the goals of tracking, one might presume that these expectations are rather homogeneous within tracks and heterogeneous between tracks. In Flanders (the northern region of Belgium), the educational system consists of four tracks. Many students start out in the most prestigious, academic track. If they fail to gain the necessary credentials, they move to the less esteemed technical and vocational tracks. Therefore, the educational system has been called a 'cascade system'. We presume that this cascade system creates homogeneous expectations in the academic track, though heterogeneous expectations in the technical and vocational tracks. We use data from the International Study of City Youth (ISCY), gathered during the 2013-2014 school year from 2354 pupils of the tenth grade across 30 secondary schools in the city of Ghent, Flanders. Preliminary results suggest that the technical and vocational tracks show more heterogeneity in student’s expectations than the academic track. If tracking does not fulfill the desired goals in some tracks, tracking practices should be questioned as tracking occurs along social and ethnic lines, causing social inequality

Esa 12th Conference: Differences, Inequalities and Sociological Imagination: Abstract Book

Esa 12th Conference: Differences, Inequalities and Sociological Imagination: Abstract Boo