Prof. Dr. Michael Joswig
Technische Universität Berlin
The most basic techniques in network optimization are methods to find shortest paths between pairs of nodes in a directed graph. Classical examples include the algorithms of Dijkstra and Floyd–Warshall. These are among the core tools used, e.g., in devices which help a car driver to navigate a road network. Since efficient algorithms are available the corresponding shortest–path problems can be solved almost instantly, even on cheap hardware, and even for fairly large networks. Yet the situation for the network provider is quite different from the perspective of the network user. One reason is that the provider’s objective does not necessarily agree with the one of the user: While the individual driver might be interested in short travel times, the traffic authorities of a metropolitan city might want to, e.g., minimize the total amount of pollution. More importantly, the traffic authorities seek to achieve a system optimum, whereas the driver cares for an individual objective. Typically, in relevant cases it is next to impossible to even describe a system optimum. To help circumventing this problem, this project will focus on developing mathematical tools to assess the impact of local changes to a network a priori. Our prime application will be toward the computation of shortest paths. However, it is expected that some results can also be transferred to other network optimization problems. The optimization of networks is a central theme in combinatorial optimization. Hence the literature is abundant, and the 600 pages of the first volume of Schrijver’s monograph only form the tip of the iceberg. Modern concepts in this area include online techniques as well as robustness and randomization and dynamic algorithms. There are methods which can deal with partial or even incorrect information, and these can also be useful for analyzing modifications to a network to some extent. Further, in practice simulation plays an important role, possibly even on a microscopic level with agents modeling individual drivers. Here we propose to take a somewhat different view on this subject. We will make use of methods from polyhedral geometry to allow for addressing the relevant combinatorial optimization problems in a parameterized fashion.