Project description DEENESFRITPL Another way to look at the hardest of the hard problems Some problems can’t be solved. They are the hard of the hardest. These are termed NP-hard and cannot admit worst-case efficient algorithms. The field of parameterised complexity (PC) reveals that the nutshell of hardness often lies in particular properties of the instances called parameters. At the heart of PC lies the study of path (or cycle) problems. However, fundamental questions concerning path problems remain unanswered, and the relations between the different techniques to solve path problems are unknown. The EU-funded PARAPATH project will build a unified, deep theory to analyse parameterised path problems. The project will answer the fundamental question ‘What makes an NP-hard problem hard?’ The answer will assist in the design of efficient algorithms for wide classes of instances of NP-hard problems. Show the project objective Hide the project objective Objective 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. Fields of science natural sciencesmathematicspure mathematicsalgebranatural sciencesmathematicspure mathematicsdiscrete mathematicsgraph theorynatural sciencesmathematicspure mathematicsdiscrete mathematicscombinatorics Programme(s) HORIZON.1.1 - European Research Council (ERC) Main Programme Topic(s) ERC-2021-STG - ERC STARTING GRANTS Call for proposal ERC-2021-STG See other projects for this call Funding Scheme HORIZON-AG - HORIZON Action Grant Budget-Based Coordinator BEN-GURION UNIVERSITY OF THE NEGEV Net EU contribution € 1 499 821,00 Address . 84105 Beer sheva Israel See on map Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Other funding € 0,00