From Graph Theory to Graph Algorithms, and Back

• Main Author: Tan, Zihan
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-2022
• Subjects: Computer science
• ISBN: 9798834007982

Graphs are among the objects that have been studied in Mathematics and Computer Science for decades. Two prominent areas that are dedicated to the study of graphs are Graph Theory, which studies the structural properties of graphs, and Graph Algorithms, which studies computational aspects of combinatorial optimization problems on graphs. Since graphs arise in many areas within Mathematics, Computer Science and beyond, results and techniques from Graph Theory and Graph Algorithms also had a great impact on a wide variety of areas.In this thesis, we study two fundamental problems that are of importance to both Graph Theory and Graph Algorithms: Graph Crossing Number and Packing Low-Diameter Spanning Trees. In the Graph Crossing Number problem, we are given a graph and are asked to find a drawing of it in the plane that minimizes the number of crossings. For low-degree graphs, almost all previous algorithms followed the same framework, and the best of them achieved an O(√n)-approximation, which was also proved to be optimal in this framework. We propose a new framework that overcomes this barrier and reduces the problem to another related problem called Crossing Number with Rotation System. The reduction relies on an algorithm for decomposing a graph into subgraphs with specific structural properties.For the Packing of Low-Diameter Spanning Trees problem, the celebrated tree-packing theorem due to Tutte and Nash-Williams states that every 2k-edge-connected graph contains k edge-disjoint spanning trees. We extend the Tutte-Nash-Williams Theorem to packing low-diameter trees into low-diameter graphs, as the low diameter property of trees/graphs is preferable in many settings (e.g., distributed algorithms). Specifically, we design a randomized algorithm to show that every 2k-edge-connected graph with diameter D contains k spanning trees with diameter kO(D) each, that cause edge-congestion at most 2. We also show that the same algorithmic techniques can be applied to improve Karger's edge-sampling theorem.In the above results, we exploit insights into the structure of graphs to design algorithms for Graph Crossing Number, and use algorithmic techniques to prove graph-theoretic results regarding packing low-diameter spanning trees, thereby strengthening the connection between Graph Theory and Graph Algorithms.