Advanced Graph Algorithms and Optimization, Spring 2025
- Lecturers: Rasmus Kyng and Max Probst with guest lectures by Tianyi Zhang
- Teaching Assistants: Simon Meierhans, Timon Kick, Nicola Widmer, Andor Vári-Kakas, Florian Trummer
- Time and Place: Mondays 10:00-12:00 at ML F 39 (always) and Tuesdays 16:00-18:00 at CAB G 61 (every other week, see schedule)
- Exercise Session Time and Place: Fridays 14:00-16:00 at CAB G 51
- ECTS credits: 10 credits
- Final deregistration date: April 10th
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Course Objective: The course will take students on a deep dive into modern approaches to graph algorithms using convex optimization techniques. By studying convex optimization through the lens of graph algorithms, students should develop a deeper understanding of fundamental phenomena in optimization. The course will cover some traditional discrete approaches to various graph problems, especially flow problems, and then contrast these approaches with modern, asymptotically faster methods based on combining convex optimization with spectral and combinatorial graph theory.
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Course Content: Students should leave the course understanding key concepts in optimization such as first and second-order optimization, convex duality, multiplicative weights and dual-based methods, acceleration, preconditioning, and non-Euclidean optimization. Students will also be familiarized with central techniques in the development of graph algorithms in the past 15 years, including graph decomposition techniques, sparsification, oblivious routing, and spectral and combinatorial preconditioning.
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Prerequisites: This course is targeted toward masters and doctoral students with an interest in theoretical computer science. Students should be comfortable with design and analysis of algorithms, probability, and linear algebra. Having passed the course Algorithms, Probability, and Computing (APC) is highly recommended, but not formally required. If you are not sure whether you are ready for this class or not, please consult the lecturer.
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Literature:
- Convex Optimization
by Stephen Boyd and Lieven Vandenberghe.
- This book is a helpful reference for convex optimization.
- Differential Calculus by Henri Cartan.
- If you have a powerful urge to learn all about multivariate calculus, you can start here. You shouldn't need it for the course though.
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Convex Analysis by Rockafellar.
- This is the classic text on basic convex analysis (not optimization). Again, you shouldn't need this.
- Convex Optimization
by Stephen Boyd and Lieven Vandenberghe.
Course Moodle
You can ask the course TAs and instructors about the course materials, exercises, and any other issues on the course Moodle page (link).
Please make sure you receive Moodle notifications from this course as some announcements will be made exclusively through Moodle.
Notes
The notes are available here (link).
These notes will be updated throughout the course. There may sometimes be a delay between a lecture being held and the corresponding Chpt. being added to the notes.
You can find the notes for an earlier version of the course here: AGAO 2024. Note that some material may change this year.
Performance Assessment
Graded Homework 1 (out approx. April 2nd, due April 16th): 15% of the grade. The homework consists of exercises.
Graded Homework 2 (out approx. May 14th, due June 4th): 15% of the grade. The homework consists of exercises.
Oral Exam (May 26th, 27th & 28th): 70% of the grade. The exam will last 15 minutes. We will mainly discuss topics from the lectures, but may ask questions about problems from the first graded homework assignment.
Bonus points (weekly exercises): Bonus up to 5% of the grade. A subset of the weekly exercises are marked with bonus points. Answering all these correctly will add points corresponding to 5 % of the total points in the course, but not answering them will not give a penalty. This bonus may increase your grade by up to 0.25. You can still get a 6 without getting any bonus points. In total, there will be 10 exercise sheets. If you hand in more than 8 exercise sheets, we consider the best 8 submissions to determine the bonus.
Graded Homework
Each graded homework assignment accounts for 15% of your grade. You should submit your solutions as a PDF typeset in LaTeX by the listed due date, 23:00 local time in Zurich, by submitting them through the course Moodle page. You must come up with the solutions on your own and write everything yourself.
Exercises
Every week we will publish a problem set consisting of exercises designed to help you understand
the course
material.
The exercises will not count toward your grade, but you are highly encouraged to solve them all.
Exercises will be published on Wednesdays on the course Moodle. You have until Thursday the following week to hand
in you solution.
The TAs will discuss solutions to the exercises one and a half weeks after they were initially published.
Part of the exercise session will be reserved for working on the current exercise sheet (published 2 days before).
The TAs will offer hints and support when needed and you're welcome to form and work in groups.
If you wish to get feedback on your problem set solutions, you must submit solutions either as a PDF typeset in
LaTeX
or in readable handwritten form
by Thursday, 23:59 local time in Zurich, by submitting them through the course Moodle page.
You're welcome to work with your classmates on these problem sets!
Bonus points from weekly exercises: To get bonus points from weekly exercises, you have to solve the particular exercises marked with bonus points. These are always clearly labelled. To get points, you must turn in your exercise sheet on time, as per the deadline above.
Tentative Schedule of Lectures and Exercises
| Date | Lecture | Topic | Problem Sets and Graded Homework |
|---|---|---|---|
| 02/17 Mon | 1 | Introduction | |
| 02/18 Tue | 2 | Basic Optimization and Linear Algebra | |
| 02/24 Mon | 3 | Convexity, Second Derivatives, Gradient Descent | |
| 02/25 Tue | no lecture | ||
| 03/03 Mon | 4 | Acceleration | |
| 03/04 Tue | 5 | Spectral Graph Theory | |
| 03/10 Mon | 6 | Effective Resistances, Gaussian Elimination as Optimization | |
| 03/11 Tue | no lecture | ||
| 03/17 Mon | 7 | Cheeger's Inequality and Introduction to Expanders | |
| 03/18 Tue | 8 | Random walks | |
| 03/24 Mon | 9 | Introduction to Random Matrices | |
| 03/25 Tue | no lecture | ||
| 03/31 Mon | 10 | Additive Gaussian Elimination and Introduction to Random Matrices | |
| 04/01 Tue | 11 | Solving Laplacian Linear Systems | |
| 04/07 Mon | 12 | Classical Maxflow 1 | |
| 04/08 Tue | no lecture | ||
| 04/14 Mon | 13 | Classical Maxflow 2 | |
| 04/15 Tue | 14 | Expanders and Sparsest Cut via the Cut-Matching Framework | |
| 04/21 Mon | EASTER | ||
| 04/22 Tue | EASTER | ||
| 04/28 Mon | 17 | Fenchel conjugates and Newton's method | |
| 04/29 Tue | no lecture | ||
| 05/05 Mon | 18 | IPM for Maximum Flow 1 | |
| 05/06 Tue | 19 | IPM for Maximum Flow 2 | |
| 05/12 Mon | 20 | Low-Diameter Decompositions | |
| 05/13 Tue | no lecture | ||
| 05/19 Mon | 21 | Negative SSSP | |
| 05/20 Tue | 22 | Graph Matchings | 05/23 Fri | 30 | Course Review | 05/26 Mon | EXAMS | 05/27 Tue | EXAMS | 05/28 Weds | EXAMS |