Loading...

Approximation Algorithms for Finding Minimum Power Dominating Sets

Ramezani, Mahdi | 2019

761 Viewed
  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 52116 (19)
  4. University: Sharif University of Technology
  5. Department: Computer Engineering
  6. Advisor(s): Zarrabi-Zadeh, Hamid
  7. Abstract:
  8. Power dominating set is a concept in graph theory that was first defined as a result of studying the controllability of electric power systems. Assume that a graph G and a subset S of vertices of G are given. First, we color all vertices in S black, and all other vertices of G white. Then we color all vertices that have a neighbor in S black (Domination step). After that, for each black vertex v, if all neighbors of v except one (the vertex u) are black, then we also color the vertex u black (Propagation step). If after a number of Propagation steps all vertices of G are black, then we call S a power dominating set of G. The minimum cardinality of a power dominating set of G is called the power domination number of G. Another graph parameter is the zero forcing number, which was defined as a result of studying the maximum nullity of graphs and controllability of quantum systems. S is a zero forcing set of G, if after some Propagation steps (without the Domination step) all vertices of G are colored black.The minimum cardinality of a zero forcing set of G is called the zero forcing number of G. The concepts of power domination and zero forcing have been generalized to k-power domination and k-forcing, respectively. In the Propagation step of these generalized concepts, if a vertex v has k or less white neighbors, then all of its neighbors can be colored black. In this project, we study the k-power domination and k-forcing numbers of some classes of grids. First, we present a method that can be used to bound the power domination number of graphs with bounded maximum degree from below. Using this method, we present an approximation algorithm with the approximation factor of 1:5 for finding a minimum power dominating set in triangular grids with rectangular border and an approximation algorithm with the asymptotic approximation ratio of 1:2 for the same problem in 3D grids.Furthermore, we provide upper and lower bounds on the pathwidth of triangular grids with triangular border and find the exact value of the zero forcing number of triangular grids with rectangular and hexagonal borders. For k ⩾ 2, the k-power domination and k-forcing numbers of triangular grids with triangular, rectangular, and hexagonal borders, square grids, and 3D grids are determined.Upper bounds on the zero forcing and power domination numbers of d-dimensional grids are presented at the end of this thesis
  9. Keywords:
  10. Power Dominating Set ; Zero-Forcing Sets ; Grid Graphs ; Approximate Algorithm ; Graph Theory

 Digital Object List