The desire for optimization not only pervades our private and working lives in countless ways, but also plays a central role in practically all areas of applied mathematics. However, the specific problems and solution methods differ greatly depending on the mathematical discipline.

Our central area of research and teaching is discrete optimization. This deals with problems whose solutions require “discrete” decisions. This contrasts with continuous optimization: for the optimal composition of a stock portfolio, for example, it may make sense to invest 52.7% of the available funds in stock A, 32.3% in stock B and 15% in stock C. In contrast, any algorithm used for a navigation system must make a full decision between discrete options at every intersection - no user of this navigation system will be happy with the suggestion to turn left at 52.7%.

Mathematically, discrete decisions can often be formulated as integer constraints. These change the properties and complexity of the optimization problems considerably, which means that the most successful ideas of continuous optimization (derivative/gradient, local extrema, convexity) are only rarely and indirectly applicable in discrete optimization.

Finite graphs or networks are a versatile tool for modeling discrete optimization problems.Therefore, our second focus is on graph theory. In addition to algorithmic aspects, we study structural properties, where there is often fruitful interaction - deep structural insights enable efficient algorithms and, conversely, algorithmic questions motivate interesting structural research. Structural questions that drive us are e.g:

Which structures characterize a certain class of graphs?
Which properties allow us to color a graph with a few colors?
How do structural parameters influence algebraic parameters of a graph?

Compared to classical fields of mathematics, discrete optimization and graph theory are comparatively young disciplines.In fact, the development of computers has contributed significantly to their dynamic development in recent decades.This is due to the fact that computers are ultimately machines for manipulating discrete data and the algorithmic methods developed and analyzed by discrete optimization are only of great practical use when implemented on computers.Due to their young age, discrete optimization and graph theory are still full of fundamental open problems that are often very easy to formulate.
It is already possible to deal with these open questions and make your own contributions as part of a Bachelor's or Master's thesis. As a result, it is not uncommon for theses to lead to scientific publications.

The proximity to relevant applications also makes the Institute of Optimization and Operations Research a suitable partner for the technical supervision of work placements. During such internships, the methods from the lecture are often further developed and adapted in order to solve specific algorithmic problems from practice.