New Tools in Parameterized Complexity: Paths, Cuts, and Decomposition (Dagstuhl Seminar 24411)

Abstract

The Dagstuhl Seminar concentrated on the development of new tools arising from the parameterized complexity of cuts, paths, and decompositions in graphs. The last 2 years were very exciting for the area, with a number of breakthroughs.
In FOCS 2021, Korhonen introduced a new method for approximating tree decompositions in graphs. His method, which was deeply rooted in classical graph theory, appeared to be a very handy tool for decomposing graphs, and several STOC/FOCS papers developed this method in various settings. In parallel, a novel perspective on graph decompositions was proposed by Bonnet et al. in FOCS 2020. The new theory of twin-width had many exciting consequences, and we were still at the beginning of understanding the real impact of the new decompositions on graph algorithms.
In a series of papers (SODA 2021, STOC 2022, SODA 2023), Kim et al. developed beautiful algorithmic methods for handling separators in (undirected, weighted, or directed) graphs by the addition of arcs. The new algorithmic tool was used to resolve a number of long-standing open problems in the area, and it also seemed to pave the road to many more new discoveries.
Reis and Rothvoss (Arxiv 2023) announced a ((log n)^{O(n)}) time randomized algorithm to solve integer programs in n variables. This breakthrough had an impact on many problems in parameterized complexity, especially on problems concerning cuts in graphs. Finally, by employing algebraic methods (both new and old), significant progress was made on several problems related to paths, including the classical (k)-disjoint path problems.
This seminar brought together people from the parameterized complexity community, specialists in cuts, flows, and connectivity, and those who had been at the forefront of these new developments. In doing so, it consolidated the results achieved in recent years, discussed future research directions, and further explored the potential applications of the methods and techniques described above.