מירב זהבי

In the Disjoint Paths problem, the input consists of an n-vertex graph G and a collection of k vertex pairs, (formula presented), and the objective is to determine whether there exists a collection (formula presented) of k pairwise vertex-disjoint paths in G where the end-vertices of (formula presented) are (formula presented) and (formula presented). This problem was shown to admit an (formula presented)-time algorithm by Robertson and Seymour Graph Minors XIII, The Disjoint Paths Problem, JCTB. In modern terminology, this means that Disjoint Paths is fixed parameter tractable (FPT) with respect to k. Remarkably, the above algorithm for Disjoint Paths is a cornerstone of the entire Graph Minors Theory, and conceptually vital to the (formula presented)-time algorithm for Minor Testing (given two undirected graphs, G and H on n and k vertices, respectively, determine whether G contains H as a minor). In this semi-survey, we will first give an exposition of the Graph Minors Theory with emphasis on efficiency from the viewpoint of Parameterized Complexity. Secondly, we will review the state of the art with respect to the Disjoint Paths and Planar Disjoint Paths problems. Lastly, we will discuss the main ideas behind a new algorithm that combines treewidth reduction and an algebraic approach to solve Planar Disjoint Paths in time (formula presented) (for undirected graphs).