====== Recent Topics in Parameterized Complexity ======

This is a three-part lecture taught by Cornelius Brand, [[https://sites.google.com/site/aefeldmann/|Andreas Emil Feldmann]] and myself.

Topics will be roughly:

  - Parameterized algorithms for integer programming and their applications in scheduling. //(Martin)//
  - Hard graph problems (e.g., longest path, longest cycle) via algebraic methods; this will be an elementary exposition, so everyone is welcome. //(Cornelius)//
  - 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)//

===== Parameterized algorithms for IP and applications =====

**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 [[https://lara.epfl.ch/w/_media/papadimitriou81complexityintegerprogramming.pdf|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 [[https://dl.acm.org/doi/10.1145/3340322|new algorithm of Eisenbrand and Weismantel]] ([[https://arxiv.org/abs/1707.00481|arxiv]]) which improves the runtime to roughly $(m \Delta)^{O(m)}$; also see an [[https://drops.dagstuhl.de/opus/volltexte/2018/10136/pdf/LIPIcs-ITCS-2019-43.pdf|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 [[https://link.springer.com/article/10.1007%2Fs10951-017-0550-0|Knop and Koutecký]] ([[https://arxiv.org/abs/1603.02611|arxiv]]).

**Second & Third lecture:** We talked about Huge $n$-fold Integer Programming, solving its Configuration LP and that a solution derived from it satisfies stronger proximity bounds than the solution of a normal relaxation. This is based on [[https://arxiv.org/abs/1909.07326|MIMO]], Section 3. I did not go into the full detail and you are not required to understand all of the proofs, but please try to understand the main concepts and statements, which are:

  * Definition of the Huge $n$-fold IP problem
  * The Configuration LP of the Huge $n$-fold IP (equations (3)-(4))
  * How the Configuration LP is solved in FPT time by solving its Dual by the Ellipsoid method (separation oracle) (Lemma 12)
  * The definition of a configurable cycle (pg. 14) and some of its properties: Lemmas 13-15
  * The proximity theorem (Thm 21) and how it is used to obtain an FPT algorithm for Huge $n$-fold IP (Proof of Thm 5)

It would be also great if you had an idea of how the proximity theorem is used to obtain [[https://arxiv.org/abs/2003.02187|kernels for scheduling problems.]]

**Fourth lecture:** We talked about Mixed Integer Linear Programs, [[https://cunicz-my.sharepoint.com/:o:/g/personal/63735541_cuni_cz/EgHZ50r_8PVJmdyiKhlSnsoBr1koI1loAthyZ2kTeyjsRQ?e=ZF5QCa|notes are here]]. In particular note we discussed a neat trick of Hochbaum-Shantikumar on how to get a Separable Convex IP algorithm from a Linear IP algorithm.
===== Parameterized Approximations for Steiner Trees =====

Covered topics and references:

  - Lossy kernels: sections 1 and 2 of [[https://arxiv.org/abs/1604.04111|Lokshanov et al.]]
  - PSAKS for Steiner Tree parameterized by nr of terminals: section 6.2 of [[https://arxiv.org/abs/1604.04111|Lokshanov et al.]]
  - Proof of Borchers and Du Theorem: sections 1 and 3 of [[https://epubs.siam.org/doi/pdf/10.1137/S0097539795281086|Borchers and Du]]
  - Directed Steiner Tree parameterized by nr of Steiner vertices: sections 1, 4, and 5 of [[https://arxiv.org/pdf/1710.00668.pdf|Dovřak et al.]]
  - Undirected Steiner Tree (and Steiner Forest) parameterized by nr of Steiner vertices (and nr of connected components): sections 1 and 3 of [[https://arxiv.org/pdf/1710.00668.pdf|Dvořak et al.]]