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Lecture starts at 16:00Written on 17.04.24 by Karl Bringmann Dear students, As decided in the lecture yesterday, from now on the lecture will always start at 16:00 sharp. |
Fine-Grained Complexity Theory
Complexity theory traditionally distinguishes whether a problem can be solved in polynomial time (by providing an efficient algorithm) or the problem is NP-hard (by providing a reduction). However, for practical purposes the label "polynomial-time" is too coarse: It can make a huge difference whether an algorithm runs in say linear, quadratic, or cubic time. In this course we explore an emerging subfield at the intersection of complexity theory and algorithm design which aims at a more fine-grained view of the complexity of polynomial-time problems. We present a mix of algorithms and conditional lower bounds for fundamental problems, often by drawing interesting connections between seemingly unrelated problems. A prototypical result presented in this course is the following: If there is a substantially faster algorithm for computing all-pairs shortest paths in a weighted graph, then there also is a substantially faster algorithm for checking whether the graph has a negative triangle (and vice versa). The techniques for proving such statements have been developed relatively recently and the majority of the results taught in this course are less than ten years old.
Time & Date & Format
This course consists of one lecture per week (Tuesday 16:00-17:30) and a tutorial every other week (Friday 10:15-12:00).
The first lecture is on April 16. The first tutorial is held on April 26, then starting from May 03 a tutorial is held in every second week. Lectures and tutorials are held physically in room 024 building E1 4.
Prerequisites
We assume basic knowledge in algorithms and theoretical computer science, as taught in the basic courses "Grundzüge von Algorithmen und Datenstrukturen"/"Fundamentals of Algorithms and Data Structures" and "Grundzüge der Theoretischen Informatik"/"Introduction to Theoretical Computer Science". The core lecture "Algorithms and Data Structures" is useful, but no formal prerequisite.
Registration
You need to register on this webpage to get access to exercise sheets and other course material.
Exercises
An important part of the course are the exercises, where you will design conditional lower bounds essentially on your own. There will be 6 exercise sheets and you need to collect at least 50% of all points on exercise sheets to be admitted to the exam. You are allowed to collaborate on the exercise sheets, but you have to write down a solution on your own, using your own words. Please indicate the names of your collaborators for each exercise you solve. Further, cite all external sources that you use (books, websites, research papers, etc.).
Exam
To be admitted to the exam, you need to achieve 50% of the points on the exercise sheets. There will be a written or oral exam.
Schedule
This is a tentative list of topics.
| Lecture | Tutorial | Topic |
|---|---|---|
| Apr 16 | Introduction | |
| Apr 23 | SAT Algorithms | |
| Apr 26 | Q&A and presence exercises | |
| Apr 30 | Lower Bounds from P!=NP and ETH | |
| May 3 | Tutorial | |
| May 07 | Lower Bounds from SETH and OVH: Hardness of LCS | |
| May 14 | Subcubic Equivalences of APSP | |
| May 17 | Tutorial | |
| May 21 | Subcubic Equivalences of APSP continued | |
| May 24 | Irregular Lecture Slot 10:15 - Matrix Multiplication and special cases of APSP | |
| May 28 | Lower bounds from 3SUM | |
| May 31 | Tutorial | |
| Jun 04 | Cancelled | |
| Jun 11 | More lower bounds from 3SUM | |
| Jun 14 | Tutorial | |
| Jun 18 | Counting to Decision, Hardness of Approximation in P | |
| Jun 25 | Cancelled | |
| Jun 28 | Tutorial | |
| Jul 02 | Subset Sum lower bound from SETH | |
| Jul 05 | Irregular Lecture Slot 10:15 - Subset Sum algorithms | |
| Jul 09 | Cancelled | |
| Jul 12 | Tutorial | |
| Jul 16 | MinPlusConvolution, Knapsack | |
| Jul 19 | Irregular Lecture Slot 10:15 - Clique and Hyperclique | |
| Jul 23 | Outlook, Lower Bounds for Dynamic Problems | |
| Jul 26 | Tutorial |
Literature
We do not follow a particular book. You can find slides and partial lecture notes from previous iterations of this course on the following webpages. We closely follow the course from 2021.
Winter 2021: Fine-Grained Complexity Theory
Summer 2019: Fine-Grained Complexity Theory
Summer 2018: Selected Topics in Fine-Grained Complexity Theory