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Conditional Geometric Touringand Connectivity

Ahadi, Arash | 2018

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 51089 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Zarei, Alireza
  7. Abstract:
  8. Finding optimize tours on a given sequence of objects has applications in robitic. A tour on a given sequence of objects is a path that touchs or cuts each of them, in order. In STOC′03 it is shown that finding such a shortest path for a sequence of convex polygons is polynomial solvable and it is NP-hard for non-convex polygons with intersections. The complexity of the problem for disjoint polygons is asked as the importest open peoblem. In 2008 an approximation algorithm is presented for this problem. We show that the problem is NP-hard in each Lp norm, even if each polygon consists of two unit line segments. Also, in 2003 the problem, with obstacles has been proposed as a future work. An obstacle is a polygon that the tour cannot enters its inside. We present a polynomial time algorithm for this problem. In this decade, researchers introduced similar problems on uncertain data; that can be considered as a generalization of the above problem. In such a problem, the input is a graph G such that its vertices are some polygons in the plane. Each polygon is the uncertain location of a point. The goal is finding a shortest path between two given points or finding the minimum spanning tree of G. The worst case of the minimum spanning tree means selecting one point from each polygon, such that the minimum spanning tree of G has the maximum possible lenght. The complexity of this problem is asked in 2008. We show its NP-hardness. Another similar problem is minimizing the number of new guards in order to connect the visibility graph of a given set of guards in a given simple polygon. Each guard is a point and in the visibility graph of guards, two guards are connected, if see each other. This problem is introduced in 2010 and its complexity is asked. We show the NP-hardness and present a constant factor approximation algorithm
  9. Keywords:
  10. Art Gallery Problem ; Uncertain Data ; Computational Geometry ; Spanning Tree ; Touring Polygons ; Connectiny Gaurds Problem

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