Project description
Boosting research in graph theory
Graph theory deals with the study of graphs which are fundamental to modelling pairwise relations between objects. Graphs can be used to model many types of relations in physical, biological, social and information systems. Funded by the Marie Skłodowska-Curie Actions programme, the CoSP project will merge expertise in discrete mathematics with theoretical computer science to study a number of interesting topics in graph theory. These include the matching theory for graphs and hypergraphs, complex algorithms, colouring problems and graph homomorphisms.
Objective
The project brings together combinatorialists of various fields with the aim that they will enrich each other’s techniques. The tool kits they will bring include topology, probability, statistical physics and algebra. These should apply to matching problems (a central topic in combinatorics), algorithmic problems, coloring problems (which are decompositions into independent sets or matchings) and homomorphisms (a generalization of colorings).
One umbrella under which many of these can be gathered is the intersection of two matroids, a notion generalizing that of matchings in bipartite graphs. Researchers are baffled by a strange phenomenon – that moving from one matroid to the intersection of two matroids sometimes costs little. The algorithmic problems are indeed harder, but the difference between min and max in the min-max theorems suffer only a conjectured penalty of 1.
This connects with a second direction of the research, fine grained complexity, which deals with polynomially solvable problems, and aims to prove, under widely believed assumptions, lower bounds on the exponents in the polynomial bounds. A major question in the field is proving similar tight bounds for approximation problems.
A direction connecting matchings, colorings and homomorphisms was initiated recently in statistical physics. It investigates typical algorithmic complexity, of computational problems taken under some probability distribution. While the worst case complexity questions are difficult in general and not clearly practically relevant, when we restrict to a given probability distribution of instances and when we are interested in high probability results, progress has been made, that has contributed also algorithmic insights beyond the probabilistic setting. We propose to address several outstanding open questions from the field.
Finally we will work on a deep connection, studied by some of the researchers in the project, between Ramsey theory, Model theory and graph homomorphisms.
Fields of science
Not validated
Not validated
- natural sciencesmathematicspure mathematicsdiscrete mathematicsmathematical logic
- natural sciencesmathematicspure mathematicstopology
- natural sciencesmathematicspure mathematicsalgebra
- natural sciencesmathematicspure mathematicsdiscrete mathematicsgraph theory
- natural sciencesmathematicspure mathematicsdiscrete mathematicscombinatorics
Keywords
Programme(s)
Coordinator
116 36 Praha 1
Czechia