Fine-Grained Complexity Theory

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
May 07		Lower Bounds from SETH and OVH
May 14		Subset Sum lower bound from SETH
	May 17
May 21		Subset Sum algorithms
May 28		Subcubic Equivalences of APSP
	May 31
Jun 04		Subcubic Equivalences of APSP continued
Jun 11		Matrix Multiplication and special cases of APSP
	Jun 14
Jun 18		Lower bounds from 3SUM
Jun 25		More lower bounds from 3SUM
	Jun 28
Jul 02		Lower Bounds for Dynamic Problems
Jul 09		Hardness of Approximation in P
	Jul 12
Jul 16		?
Jul 23		Outlook (recap, further hypotheses, multivariate lower bounds)
	Jul 26

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

Winter 2017: Fine-Grained Complexity Theory

Summer 2016: Complexity Theory of Polynomial-Time Problems