Skip to main content
Go to the home page of the European Commission (opens in new window)
English English
CORDIS - EU research results
CORDIS

Parameterized Complexity Through the Lens of Path Problems

Periodic Reporting for period 1 - PARAPATH (Parameterized Complexity Through the Lens of Path Problems)

Reporting period: 2022-07-01 to 2024-12-31

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) 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.
So far, we have proved several central results - some of these results resolve goals stated in the project, while other results constitute substantial progress towards the resolution of other goals and the understanding of the proposal's topic in general. For the sake of illustration, we will describe five of these results here:

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.
So far, the results we have proved already led to a fundamentally deeper understanding of the two most important path problems on which the proposal focuses: Disjoint Paths and Long Path. In particular, we now have a clear picture of Disjoint Paths on planar graphs and a better understanding of above-guarantee parameterizations of path problems. This, in turn, effect our knowledge of graph minors in general. Other byproducts of our work are also the enrichment of tools to analyze and develop kernels in general, temporal graphs in general, and algorithms for solving graph drawing problems in general. Our results have impact on Parameterized Complexity and algorithm design in general, graph theory with emphasis on graph minors, and computational geometry with emphasis on graph drawing and geometric graphs. As a side note, the publications describing these results - although they have just become available - have already been cited a significant number of times overall.
My booklet 0 0