## Periodic Reporting for period 1 - DISTRUCT (Structure Theory for Directed Graphs)

**Reporting period:**2015-07-01

**to**2016-12-31

## Summary of the context and overall objectives of the project

"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 occur frequently in practice.

A particularly rich class of these problems can elegantly be formalised using the concepts of graphs. 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 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, for instance 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 the real of 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."

A particularly rich class of these problems can elegantly be formalised using the concepts of graphs. 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 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, for instance 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 the real of 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

Work so far has concentrated on the first three work packages, WP1 - WP3.

In WP1 we proposed to work on the directed excluded grid theorem, directe disjoint paths and the Erdos-Posa property. In this work package we made very significant progress on all three objectives. For the Erdos-Posa property we provided a complete characterisation of those strongly connected digraphs that have the EP property. For digraphs that are not strongly connected we studied the class of vertex cyclic digraphs and we have achieved nearly a complete characterisation for these as well. There are a few open problems left that we hope to close before submitting it to a journal. The preprint is available from arXiv.org. (https://arxiv.org/abs/1603.02504)

For routing problems we studied routing under congestion in acyclic digraphs as a first step. The results were published at MFCS 2016 and are available at arXiv https://arxiv.org/abs/1605.01866. Furthermore, we studied the problem whether there is a polynomial time algorithm, for every fixed k, for solving the k-disjoint paths problems with congestion 2 on directed graphs. We believe that we have made significant progress and found a quasi-polynomial time algorithm. We hope to improve this to polynomial time. A preprint will be available soon.

As for directed grids, we made very good progress in this aspect as well. First of all, we significantly improved our directed excluded grid theorem and provided a much simpler proof. The new proof has been submitted to the Journal of the AMS and is currently under review. Directed grids and grid-like structures have been used by Chekuri and others for routing in directed planar digraphs. For these applications it would be a great step forward if one could show a polynomial grid theorem for planar digraphs. We have therefore worked on this problem and we think we are very close to finish it. I expect this to be done in the summer 2017. We are also intensively working on directed tree width sparsification, but the results are not ready yet.

In WP2 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. The results were presented at SODA 2017 and have been invited to the SODA best paper special issue that will appear at the ACM Transaction on Algorithms.

We also studied kernels for dominating set problems and managed to pinpoint exactly the classes of undirected graphs having polynomial kernels for these problems. This one of the first times where such a precise characterisation can be given.

As a second step we started to use these results for undirected graphs to develop the corresponding tools in the directed setting, the objective of WP2. This project has by far exceeded our expectation. We showed that directed bounded expansion classes can be defined by generalised colouring numbers, we proved that for such classes we have very efficient neighbourhood covers, the neighbourhood complexity is small and we have a splitter game that can be used for bounded depth search tree methods. The results will be presented at STACS 2017

Since submitting to STACS we have made further progress and we have now obtained polynomial kernels for dominating set problems on directed bounded expansion classes and we have been able to prove that they admit low dag-depth colourings, a tool that in the undirected world has had huge algorithmic impact. We think that this impact will translate to the directed setting also.

in WP3 we looked at combinations of density and directed tree width arguments. A first attempt was shown to be algorithmically less fruitful that we imagined. But the results we obtained in WP2 on low dag-depth colourings have shifted our attention in this WP towards digraph classes of bounded dag-depth and low density. This will be studied intensively from now on as any result we obtain here could potentially be lifted to bounded expansion classes using the low dag-depth colouring technique.

In WP1 we proposed to work on the directed excluded grid theorem, directe disjoint paths and the Erdos-Posa property. In this work package we made very significant progress on all three objectives. For the Erdos-Posa property we provided a complete characterisation of those strongly connected digraphs that have the EP property. For digraphs that are not strongly connected we studied the class of vertex cyclic digraphs and we have achieved nearly a complete characterisation for these as well. There are a few open problems left that we hope to close before submitting it to a journal. The preprint is available from arXiv.org. (https://arxiv.org/abs/1603.02504)

For routing problems we studied routing under congestion in acyclic digraphs as a first step. The results were published at MFCS 2016 and are available at arXiv https://arxiv.org/abs/1605.01866. Furthermore, we studied the problem whether there is a polynomial time algorithm, for every fixed k, for solving the k-disjoint paths problems with congestion 2 on directed graphs. We believe that we have made significant progress and found a quasi-polynomial time algorithm. We hope to improve this to polynomial time. A preprint will be available soon.

As for directed grids, we made very good progress in this aspect as well. First of all, we significantly improved our directed excluded grid theorem and provided a much simpler proof. The new proof has been submitted to the Journal of the AMS and is currently under review. Directed grids and grid-like structures have been used by Chekuri and others for routing in directed planar digraphs. For these applications it would be a great step forward if one could show a polynomial grid theorem for planar digraphs. We have therefore worked on this problem and we think we are very close to finish it. I expect this to be done in the summer 2017. We are also intensively working on directed tree width sparsification, but the results are not ready yet.

In WP2 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. The results were presented at SODA 2017 and have been invited to the SODA best paper special issue that will appear at the ACM Transaction on Algorithms.

We also studied kernels for dominating set problems and managed to pinpoint exactly the classes of undirected graphs having polynomial kernels for these problems. This one of the first times where such a precise characterisation can be given.

As a second step we started to use these results for undirected graphs to develop the corresponding tools in the directed setting, the objective of WP2. This project has by far exceeded our expectation. We showed that directed bounded expansion classes can be defined by generalised colouring numbers, we proved that for such classes we have very efficient neighbourhood covers, the neighbourhood complexity is small and we have a splitter game that can be used for bounded depth search tree methods. The results will be presented at STACS 2017

Since submitting to STACS we have made further progress and we have now obtained polynomial kernels for dominating set problems on directed bounded expansion classes and we have been able to prove that they admit low dag-depth colourings, a tool that in the undirected world has had huge algorithmic impact. We think that this impact will translate to the directed setting also.

in WP3 we looked at combinations of density and directed tree width arguments. A first attempt was shown to be algorithmically less fruitful that we imagined. But the results we obtained in WP2 on low dag-depth colourings have shifted our attention in this WP towards digraph classes of bounded dag-depth and low density. This will be studied intensively from now on as any result we obtain here could potentially be lifted to bounded expansion classes using the low dag-depth colouring technique.

## 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 most spectacular results have already been described and put in context in the work report. Here we briefly highlight the main achievements:

- using tools form model theory, more precisely stability theory, we managed to improve the running time of all algorithms using the uniformly quasi-wideness property of nowhere dense classes of undirected graphs many times exponentially. This result is applicable to a broad range of algorithmic problems on nowhere dense classes and we expect it to have broad impact.

This result far exceeded our expectations when we first started to look at stable classes of graphs.

- We showed that for classes of digraphs of bounded directed expansion we nearly have the same set of characterisations and therefore the same set of algorithmic tools available as for undirected classes of bounded expansion. This again exceeded by far our expectations when we started this project as usually translating from the undirected into the directed world does not come out with such a smooth theory. We are extremely excited about this project and believe that this will provide classes with a very elegant algorithm theory. Also, recent work by Gneis et al have shown that bounded expansion covers many networks from practical benchmarks so that these results do become applicable for practical purposes also.

- On directed grids we think we are close to a polynomial grid theorem for planar digraphs which would have very significant impact on directed routing problems as studied, e.g. by Chekuri et al.

- using tools form model theory, more precisely stability theory, we managed to improve the running time of all algorithms using the uniformly quasi-wideness property of nowhere dense classes of undirected graphs many times exponentially. This result is applicable to a broad range of algorithmic problems on nowhere dense classes and we expect it to have broad impact.

This result far exceeded our expectations when we first started to look at stable classes of graphs.

- We showed that for classes of digraphs of bounded directed expansion we nearly have the same set of characterisations and therefore the same set of algorithmic tools available as for undirected classes of bounded expansion. This again exceeded by far our expectations when we started this project as usually translating from the undirected into the directed world does not come out with such a smooth theory. We are extremely excited about this project and believe that this will provide classes with a very elegant algorithm theory. Also, recent work by Gneis et al have shown that bounded expansion covers many networks from practical benchmarks so that these results do become applicable for practical purposes also.

- On directed grids we think we are close to a polynomial grid theorem for planar digraphs which would have very significant impact on directed routing problems as studied, e.g. by Chekuri et al.