Parameterized Algorithms

Parameterized Algorithms

This course is about designing fast algorithms for NP-hard graph theoretic problems, where the running time depends on multiple parameters of the input. For example, while a database may contain a very large amount of data, the size of the database queries is typically extremely small in comparison. The aim would be to obtain algorithms that have a small dependence on the database size, but possibly a larger dependence on the query size. Such an algorithm would be fast when the queries are small. Similarly, if the goal is to find small solutions in a large graph, then an algorithm with exponential dependence on the size of the solution and polynomial dependence on the size of the graph might be acceptable.

We will see several algorithmic techniques to design fast algorithms for NP-hard problems in this setting, called Fixed-Parameter Tractable (FPT) algorithms, as well as an overview of the lower-bound methods. We will also learn about preprocessing or data-reduction algorithms in this setting, called Kernelization algorithms, which run in polynomial time and reduce a given instance of a NP-hard problem to an equivalent but much smaller instance.

Some example topics that will be covered during the course:

• Branching, bounded-depth search trees
• Randomization, color coding
• Iterative compression
• Kernelization, sunflower lemma, crown decomposition
• Kernelization lower bounds
• Algebraic methods, inclusion-exclusion
• Representative sets and matroids
• Important cuts
• Treewidth, bidimensionality on planar graphs
• Turing and lossy kernelization

Format

Two hours of lectures every week and two hours of tutorials every other week. During the semester, 5 homework exercise sheets will be handed out, to be submitted in one week. 50% of all points on the exercise sheets are needed to be admitted to the (oral) exam. The solutions of the exercises are discussed in the tutorial sessions (after the deadline). 

Lectures: Tuesday, 10:15-12:00

Tutorial: To be decided

First lecture: October 15, 2024

Room: building E1 4, room 0.24

Prerequisites

Basic knowledge of algorithms, graph theory and probability will be assumed.

Date	Topic	Material	Reference (see below)	Exercise	Due
October 15	L01: Introduction I	Slides	1, 3.1, 3.2
October 22	L02: Introduction II	Slides	1, 2.1, 2.2.1, 3.1, 3.2, 3.3, 3.5, 5.1, 5.2	Sheet 1	October 29
October 29	L03: Introduction III	Slides	6.1, 4.1, 4.2, 4.3.1, 4.4
November 5	L04: Complexity of parameterized problems	Slides	13.1, 13.2, 13.3, 13.6, 14.1, 14.2, 14.3	Sheet 2	November 12
November 12	L05: Kernelization I	Slides	2.1, 2.2, 2.6
November 19	L06: Kernelization II	Slides	2.3,2.4,2.5,3.4	Sheet 3	November 26
December 3	L07: Kernelization III	Slides	9.1, 9.4
December 10	L08: Lower Bounds for Kernelization	Slides	15.1.1, 15.1.2, 15.2.1, 15.2.2, 15.3.
December 17	L09: Important Cuts	Slides	8.1, 8.2, 8.3, 8.5, 8.6	Sheet 4

Reference Textbook

"Parameterized Algorithms" by Cygan et al. (see this for free pdf of the book from the authors).