Periodic Reporting for period 5 - LaDIST (Large Discrete Structures)
Berichtszeitraum: 2021-01-01 bis 2021-12-31
The main aim of this project has been to study and develop mathematical methods to analyze and approximate large discrete structures, in particular large graphs. A graph is a mathematical model of a network - a computer network, a social network, etc. So, the methods that have been studied in this project are of importance in particular in computer science where very large networks appear frequently.
In the framework of the project, we have analyzed mathematical objects that are used to represent large dense graphs, which are called graphons, and other large discrete structures such as permutations, Latin squares, etc. One of the main results of the project is a counterexample to a conjecture of Lovász that every problem in extremal graph theory has a finitely forcible optimum solution, which was also discussed during his Abel Prize Lecture in 2021 (a graphon is finitely forcible if it is uniquely determined by finitely many density constraints). The gained insights have been used to solve specific problems in extremal combinatorics, including several long-standing open problems. Applications of the developed methods in computer science have also been sought. While the project has not included developing software applications, some of the obtained results are algorithmic nature and have relevance to computer science. In such cases, the obtained results have been presented at appropriate computer science conferences to bring them to the attention of the computer science community.
In the framework of the project, we have analyzed mathematical objects that are used to represent large dense graphs, which are called graphons, and other large discrete structures such as permutations, Latin squares, etc. One of the main results of the project is a counterexample to a conjecture of Lovász that every problem in extremal graph theory has a finitely forcible optimum solution, which was also discussed during his Abel Prize Lecture in 2021 (a graphon is finitely forcible if it is uniquely determined by finitely many density constraints). The gained insights have been used to solve specific problems in extremal combinatorics, including several long-standing open problems. Applications of the developed methods in computer science have also been sought. While the project has not included developing software applications, some of the obtained results are algorithmic nature and have relevance to computer science. In such cases, the obtained results have been presented at appropriate computer science conferences to bring them to the attention of the computer science community.
This ERC project belongs to discrete mathematics, however, it has also addressed several topics from computer science. The studied problems have been concentrated along the following four main themes of the project: structure of dense graph limits, extremal combinatorics, bridging the dense and sparse limits, and algorithmic challenges. In all four areas, new substantial results have been obtained with most work done around the topics that belong to the first two themes. The full list of research papers prepared with the support of the project can be found at the dedicated website https://www.fi.muni.cz/research/ladist/publications.html.en. All papers supported by the project are made freely available on the preprint server arXiv at the time of submission. The project results have been a subject of 33 research papers published or accepted for publication in journals, 3 extended abstracts appeared in the proceedings of computer science conferences and 7 in the proceedings of EuroComb; additional 14 research papers are under review for publication. The results of the project were also the subject of one survey paper and another survey paper is to be finalized and submitted soon. It should be noted that the high number of papers not yet accepted for publication is caused by a slow speed of the review process in mathematics and theoretical computer science. The project also supported three research workshops that brought together researchers from different areas of discrete mathematics and computer science - the first two took place at the University of Warwick in December 2016 and in June 2018, and the last in June 2019 at the Masaryk University.
The project has gone significantly beyond the start of the art. The methods developed within the framework of the project led to solving many open problems, some open for decades. The particular highlight results include proving the universality of finitely forcible graphons, a counterexample to a conjecture of Lovász on finite forcibility of optimal solutions of extremal graph theory problems (which was the second most often quoted open problem in the theory of graph limits) and construction of common graphs with a high chromatic number (a problem opened for more than two decades). The first two of these results were also mentioned in the Abel Prize Lecture by Lovász in 2021. Applications of combinatorial methods in computer science have also been extended, for example, a new link between matroid theory and Graver bases in relevance to integer programming has been established.