Periodic Reporting for period 1 - PARAPATH (Parameterized Complexity Through the Lens of Path Problems)
Reporting period: 2022-07-01 to 2024-12-31
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) little 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 I will be able to answer decades-old questions in PC, which will revolutionize the power of this field. This includes establishing 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.
1. Planar Disjoint Paths: We proved that the problem does not admit a polynomial kernel w.r.t. k. Complementarily, we proved that it admits a polynomial kernel w.r.t. k+tw. In turn, the latter result yielded a simple FPT algorithm for the problem where the dependency on the parameter matches the best-known one. Moreover, both results together yielded that the problem does not admit a polynomial treewidth reduction.
2. We developed the currently best-known polynomial kernels for the Disjoint Paths problem on subclasses of chordal graphs.
3. We showed that Long Path Above Girth is W[1]-hard on digraphs. Towards this, we also showed that Hamiltonian Path is W[1]-hard on digraphs when parameterized by DFVS.
4. We studied the Disjoint Paths problem on temporal graphs, and obtained a wide spectrum of new results in this context.
5. We defined a new width notion based on treewidth useful for resolving graph drawing problems.