Estimating the Maximum Hidden Vertex Set in Polygons

It is known that the MAXIMUM HIDDEN VERTEX SET problem on a given simple polygon is NP-hard [11], therefore we focused on the development of approximation algorithms to tackle it. We propose four strategies to solve this problem, the first two (based on greedy constructive search) are designed specifically to solve it, and the other two are based on the general metaheuristics Simulated Annealing and Genetic Algorithms. We conclude, through experimentation, that our best approximate algorithm is the one based on the Simulated Annealing metaheuristic. The solutions obtained with it are very satisfactory in the sense that they are always close to optimal (with an approximation ratio of 1.7, for arbitrary polygons; and with an approximation ratio of 1.5, for orthogonal polygons). We, also, conclude, that on average the maximum number of hidden vertices in a simple polygon (arbitrary or orthogonal) with n vertices is n4

Minimum Vertex Guard problem for orthogonal polygons: a genetic approach

The problem of minimizing the number of guards placed on vertices needed to guard a given simple polygon (MINIMUM VERTEX GUARD problem) is NP-hard. This computational complexity opens two lines of investigation: the development of algorithms that determine approximate solutions and the determination of optimal solutions for special classes of simple polygons. In this paper we follow the first line of investigation proposing an approximation algorithm based on the general met heuristic Genetic Algorithms to solve the MINIMUM VERTEXGUARD problem

Solving the minimum vertex floodlight problem with hybrid metaheuristics

In this paper we propose four approximation algorithms (metaheuristic based), for the Minimum Vertex Floodlight Set problem. Urrutia et al. [9] solved the combinatorial problem, although it is strongly believed that the algorithmic problem is NP-hard. We conclude that, on average, the minimum number of vertex floodlights needed to illuminate a orthogonal polygon with n vertices is n/4,29

Escondiendo puntos en espirales e histogramas

El problema de maximizar el número de vértices que no son visibles dos a dos en un polígono simple P, (MAXIMUN HIDDEN VERTEX SET) es un problema NP-duro [6]. En este trabajo se resuelve el problema para dos tipos de polígonos: espirales e histogramas. Para los primeros se obtiene un algoritmo lineal que resuelve el problema MHVS y cotas para el máximo número h de vértices ocultos, [r2 ]+ 1 ≤ h ≤ r + 1, siendo r el número de vértices cóncavos del polígono espiral. Para polígonos histograma se demuestra que h = r − (p − 1), siendo p el número de lados fondo

Connecting red cells in a bichromatic Voronoi diagram

Let S be a set of n + m sites, of which n are red and have weight wR, and m are blue and weigh wB. The objective of this paper is to calculate the minimum value of wR such that the union of the red Voronoi cells in the weighted Voronoi diagram of S is a connected set. The problem is solved for the multiplicatively-weighted Voronoi diagram in O((n+m)^2 log(nm)) time and for the additively-weighted Voronoi diagram in O(nmlog(nm)) time