Graphs are fundamental objects that find widespread applications across computer science and beyond. Graph Theory has yielded deep insights about structural properties of various families of graphs, which are leveraged in the design and analysis of algorithms for graph optimization problems and other computational optimization problems. These insights have also proved helpful in understanding the limits of efficient computation by providing constructions of hard problem instances. At the same time, algorithmic tools and techniques provide a fresh perspective on graph theoretic problems, often leading to novel discoveries. In this thesis, we exploit this symbiotic relationship between graph theory and algorithms for graph optimization problems and beyond. This thesis consists of three parts. In the first part, we study a graph routing problem called the Node-Disjoint Paths (NDP) problem. Given a graph and a set of source-destination pairs of its vertices, the goal is to route the maximum number of pairs via node-disjoint paths. We come close to resolving the approximability of NDP by showing that it is $n^{\Omega(1/poly\log\log n)}$-hard to approximate, even on grid graphs, where n is the number of vertices. In the second part of this thesis, we use graph decomposition techniques developed for efficient algorithms to derive a graph theoretic result. We show that for every n-vertex expander graph G, if H is any graph with at most $O(n/\log n)$ vertices and edges, then H is a minor of G. In the last part, we show that the graph theoretic tools and graph algorithmic techniques can shed light on problems seemingly unrelated to graphs. We show that the randomized space complexity of the Longest Increasing Subsequence (LIS) problem in the streaming model is intrinsically tied to the query-complexity of the Non-Crossing Matching problem on graphs in a new model of computation that we define.

翻译：图是基础性研究对象，在计算机科学及其他领域有广泛应用。图论深入揭示了各类图族的结构特性，这些特性被用于图优化问题及其他计算优化问题的算法设计与分析。同时，这些见解还通过构造困难问题实例，有助于理解高效计算的局限性。另一方面，算法工具与技术为图论问题提供了新视角，常催生新发现。本论文利用图论与算法在图优化问题及其他领域中的共生关系展开研究，共分为三部分。第一部分研究图路由问题——节点不相交路径（NDP）问题。给定一个图及若干源-目标节点对，目标是最大化通过节点不相交路径路由成功的对数。我们证明NDP问题即使在网格图上也是$n^{\Omega(1/poly\log\log n)}$-难近似（其中n为顶点数），从而接近解决其可近似性。第二部分使用为高效算法开发的图分解技术推导图论结果：证明对任意n顶点扩展图G，若H是顶点数和边数不超过$O(n/\log n)$的图，则H是G的次要图。最后部分证明图论工具与图算法技术可阐明看似与图无关的问题：我们证明流式模型中最长递增子序列（LIS）问题的随机空间复杂度，本质上与我们定义的新型计算模型中图上非交叉匹配问题的查询复杂度相关联。