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). For practical purposes however the label "polynomial-time" is too coarse: It may 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 quite recently and most results taught in this course are less than ten years old.
Time & Date & Format
This course consists of one lecture per week (Monday 16:15-18:00) and a tutorial every other week (Friday 14:15-16:00).
The first lecture is on October 25. The lecture will be in a hybrid format, so you can join physically in room 024 building E1 4 or virtually on Zoom (the link can be found under "Information"->"Virtual Guide" after registering and logging into this website). A recording will be made available. We will discuss in the first lecture in what format the lecture should continue (physical+video / physical+Zoom / virtual-only).
We assume basic knowledge in algorithms and theoretical computer science, as taught in the introductory 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" could be useful, but is no formal prerequisite.
You need to register on this webpage to get access to exercise sheets and other course material.
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.).
To be admitted to the exam, you need to achieve 50% of the points on the exercise sheets. There will be an oral exam on February 24.
This is a tentative list of topics.
|No lecture - holiday
|Lower Bounds from P!=NP and ETH
|Lower Bounds from SETH and OVH
|Subcubic Equivalences of APSP
|Subcubic Equivalences of APSP continued
|Matrix Multiplication and special cases of APSP
|Lower bounds from 3SUM
|More lower bounds from 3SUM
|Subset Sum lower bound from SETH
|Subset Sum algorithms
|Hardness of Approximation in P
|Lower Bounds for Dynamic Problems
|Outlook (recap, further hypotheses, multivariate lower bounds)
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 (however, the content of the previous iterations somewhat differs; essentially this year's iteration is a subset of the previous ones).