Structure Theory for Directed Graphs

Project Information

DISTRUCT

Grant agreement ID: 648527

DOI

10.3030/648527

Project closed

EC signature date 7 April 2015

Start date 1 July 2015

End date 30 June 2021

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 826 773,00

EU contribution

€ 1 826 773,00

Coordinated by

TECHNISCHE UNIVERSITAT BERLIN
Germany

Periodic Reporting for period 4 - DISTRUCT (Structure Theory for Directed Graphs)

Reporting period: 2020-01-01 to 2021-06-30

Summary of the context and overall objectives of the project

Graphs are a simple mathematical model that is widely used to model real-world systems and phenomena. A graph consists of a set of objects, called vertices, and a set of edges connecting pairs of vertices. The set of vertices could be a group of people with edges representing whether two people know each other or the vertices could correspond to web pages and the edges correspond to hyperlinks between webpages.

Graphs can be distinguished between undirected and directed graphs, depending on whether the edges always go in both directions, such as in the example of people knowing each other, or can be directed only in one direction such as in the hyperlink example.

The elegance of graphs as mathematical abstraction is their simplicity, which allows to abstract from many irrelevant details of real practical problem instances.

One of the early observations in computer science was that many algorithmic problems are computationally intractable, formalised by the mathematical concept of "NP-hardness". This includes many algorithmic problems that frequently occur in practice. A particularly rich class of these problems can elegantly be formalised using the concepts of graphs.

One very successful way to overcome the computational hardness of many algorithmic problems on graphs is to study classes of graphs that have a particularly simple structure. A prominent example are planar graphs that can be drawn on a sheet of paper without edges crossing each other. It turned out that this structure can be exploited to design highly efficient algorithms for hard computational problems on input instances that are planar graphs. This has led to a very well developed theory of structural properties of graphs that can be used in the design of efficient algorithms.

However, much of this work has focussed on undirected graphs. The goal of this project is to generalise a part of this theory, known as graph minor theory and nowhere denseness, to directed graphs. In this way, the objective is to make this structural theory of graphs applicable also for algorithmic problems that are naturally modelled as directed graphs.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

One of the central objectives of the project was to generalise the graph minor structure theory for undirected graphs to the case of directed graphs.

We have made very significant progress on this part of the project. Based on the directed grid theorem which we proved earlier on we proved the next two important steps towards achieving a full structure theorem:


- We introduced the concept of directed tangles which identify highly connected parts of a digraph and proved that these highly connected parts of any digraph are arranged as a tree-structure. This provides a decomposition of any digraph into a tree of its highly connected regions which is a useful tool in a range of algorithmic applications.
- The other major achievement is the directed flat wall theorem.

On the algorithmic side of the project, we have used the combintation of these tools to develop efficient polynomial time algorithms for low congestion routing in directed graphs.

A slightly different approach to identifying beneficial substructures in graphd and digraphs is sparsity theory. We proposed to work on nowhere crownful and directed bounded expansion/directed nowhere dense classes. Here we made spectacular progress. First of all, we studied the corresponding question on the simpler case of undirected graphs. As a really surprising result we introduced a new method borrowed from stability theory, a subbranch of model theory in mathematical logic, to the study of nowhere dense classes of graphs. This way we managed to improve bounds on the uniformly quasi-wideness of such classes from many-fold exponential to polynomial. This has immediate impact on all algorithms using uniformly quasi-wideness and speeds them up many-fold exponentially. This is a really surprising result as usually tools from logic do not yield very efficient bounds.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The results of the project have pushed the

Within the project we have made very significant contributions to the field of digraph minors pushing this field far beyond the state of the art. Our results have provided a solid foundation of this field with a set of strong results that can be used in algorithmic applications.

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