This is a three-part lecture taught by Cornelius Brand, Andreas Emil Feldmann and myself.

Topics will be roughly:

1. Parameterized algorithms for integer programming and their applications in scheduling. (Martin)
2. Hard graph problems (e.g., longest path, longest cycle) via algebraic methods; this will be an elementary exposition, so everyone is wellcome. (Cornelius)
3. Parameterized approximation (i.e., how to deal with problems which are hard to approximate and do not admit fixed-parameter algorithms), in particular for network design. (Andreas)

First lecture: we talked about Integer Linear Programming with few constraints and without upper bounds, that is, the problem is $\min \mathbf{c} \mathbf{x}: \, A \mathbf{x} = \mathbf{b}, \, \mathbf{x} \geq \mathbf{0}, \, \mathbf{x} \in \mathbb{Z}^n$. First we mentioned an old algorithm of Papadimitriou with a dependence of roughly $(m \Delta)^{O(m^3)}$, where $m$ is the number of rows and $\Delta$ is the largest coefficient of $A$ in absolute value. Then, we studied a new algorithm of Eisenbrand and Weismantel (arxiv) which improves the runtime to roughly $(m \Delta)^{O(m)}$; also see an algorithm of Jansen and Rohwedder which improves the constants in the exponent. Finally, we talked about using this algorithm to get an algorithm for the scheduling problem $P||C_{\max}$ which runs in time roughly $p_{\max}^{O(p_{\max})}$ instead of the $p_{\max}^{O(p_{\max}^2)}$ time implied by Knop and Koutecký (arxiv).