Parameterized Complexity Through the Lens of Path Problems

Nowadays, numerous problems are known to be NP-hard, and hence unlikely to admit worst-case efficient algorithms. Fortunately, the field of Parameterized Complexity (PC) shows that the nutshell of hardness often lies in particular properties (called parameters) of the instances. Here, we answer the fundamental question: What makes an NP-hard problem hard? Specifically, how do different parameters of an NP-hard problem relate to its inherent difficulty? Based on this knowledge, we design efficient algorithms for wide-classes of instances of NP-hard problems. At the heart of PC lies the study of path (or cycle) problems. The inception of PC was inspired by the Graph Minors Theory, where the resolution of DISJOINT PATHS is a cornerstone. Moreover, the study of k-PATH has led to a large number of major breakthroughs in PC over the past three decades. Still, (i) fundamental questions concerning path problems have remained unanswered, and (ii) close to nothing is known about the relations between the different techniques to solve path problems. The overarching goal of this proposal is to build a unified, deep theory to analyze parameterized path problems. As known techniques to solve path problems rely, individually, on Graph Minors Theory, Extremal Combinatorics, Matroid Theory, Exterior Algebra, and more, I will draw new deep connections between these fields (towards unification). Based on the new theory, I believe that I will be able to answer decades-old questions in PC, which will revolutionize the power of this field. This includes the establishment of an Efficient Graph Minors Theory, an optimality program for color-coding-amenable problems, and a machinery to refute the existence of polynomial Turing kernels. Answers to these questions will substantially reshape the future of the design of parameterized algorithms, graph algorithms, and preprocessing procedures. Additionally, they will have high impact applications in practice.