Monitoring on Graphs: An Exploration into k-Cores, Zero Forcing, and Power Domination
Mikesell, Derek Justin
Hicks, Illya V
Doctor of Philosophy
This thesis will cover various computational models for approximating and solving multiple graph monitoring problems. The first problem of interest is the Minimum k-core problem, which asks for the smallest induced subgraph of minimum degree k. It has been shown that this problem is difficult and thus sophisticated techniques are required to obtain good solutions. The minimum k-core problem is modeled as a binary integer program and relaxed as a linear program. Since the relaxation may yield a non-integral solution, a Branch-and-Cut framework is used to find the integral optimal solution. It is shown that the edge and cycle transversals of the graph give valid inequalities for the convex hull of the k-core polytope - which can be further generalized to a family of l-core transversals. Further, a heuristic for the hitting set of the minimal l-cores is given with its associated valid inequality. Additionally, improved valid inequalities are generated using bounds generated via the girth of the graph. Multiple low order heuristics are explored for finding initial bounds for the branching process utilizing the degree distribution of the graph. Finally, numerical results are given comparing the Branch-and-Bound, Branch-and-Cut, and heuristic techniques. The next problem that is studied is the Zero Forcing Problem. Zero forcing is a graph coloring process based on the following color change rule: all vertices of a graph G are initially colored either blue or white; in each time-step, a white vertex turns blue if it is the only white neighbor of some blue vertex. A zero forcing set of G is a set of blue vertices such that all vertices eventually become blue after iteratively applying the color change rule. The zero forcing number Z(G) is the cardinality of a minimum zero forcing set. This thesis proposes novel exact algorithms for computing Z(G) based on formulating the zero forcing problem as a two-stage Boolean satisfiability problem with clause generation. Several heuristics are proposed for zero forcing based on iteratively adding blue vertices which color a large part of the remaining white vertices. These heuristics are used to speed up the exact algorithms, and can also be of independent interest in approximating Z(G). Computational results on various types of graphs show that in many cases, the algorithms presented offer a significant improvement on the state-of-the-art algorithms for zero forcing. Lastly, the Connected Power Domination problem is studied. The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network, while monitoring the entire network. A power dominating set of a graph is a set of vertices from which every vertex in the graph can be observed, following a set of rules for power system monitoring. This thesis studies the problem of finding a minimum power dominating set which is connected; the cardinality of such a set is called the connected power domination number of the graph. It is shown that the connected power domination number of a graph is NP-hard to compute in general, but can be computed in linear time in cactus graphs and block graphs. Various structural results about connected power domination are given. Finally, novel integer programming formulations for power domination, connected power domination, and power propagation time are presented, with their respective computational results showing they perform better than other integer programming models in the literature.