Combinatorial Optimization 2 (MAT.482/MAT.483)
Summer Term 2021
Contents of the course
Due to COVID19 the course this term is held in an online manner. For details see the TC course. The lectures are recorded and you can watch the videos at
your favourite time. There will be online meetings for the practical. Information will be sent to the registered students via TUG-Online.
This term there are no regular office hours.
If you happen to have questions, feel free to send me an e-mail to
I can also answer questions in the forum of the TC course or
offer a chat session if there is a demand.
Contents and Goal of the Course
The course builds on the Combinatorial Optimization 1 course which mainly focused on polynomially solvable combinatorial optimization problems (minimum spanning tree, branchings, shortest path, maximum flow, minimum cost flow, cardinatity matching, total unimodularity).
The Combinatorial Optimization 2 course is an extension of the Combinatorial Optimization 1 course and deals with important combinatorial optimization problems which are not covered in the Combinatorial Optimization 1 course with a particular focus on NP-hard combinatorial optimization problems.
The goal of the course is not only to deal with additional classes of combinatorial optimization problems, but also provide the students with an arsenal of diverse approaches to attack hard combinatorial optimization problems (exact approaches, heuristics, approximation algorithms, special cases).
I plan to cover the following topics:
- Short introduction to approximation algorithms (notions: constant factor
approximation, PTAS, FPTAS)
Covering and packing problems: bin packing, knapsack problem, heuristics, approximation algorithms+schemes, dynamic programming.
- Travelling salesman problem (TSP): lower bounds, heuristics, approximation algorithms, branch&bound, TSP polytope
- Quadratic assignment problem (QAP): lower bounds, heuristics
- Weighted nonbipartite matching problems, weighted T-joins
- Matroid intersection (if time permits, not likely to happen)
The course is not based on a text book.
In the following some recommendable books are listed.
General text books and monographs
The following four books provide a good overview about the area of combinatorial optimization.
W.J. Cook, W.H. Cunningham, W.H. Pulleyblank, A. Schrijver,
Wiley-Interscience, John Wiley & Sons, New York, 1998, 355 pages,
(Easily readable, but yet modern text book. Covers less topics than the
book by Korte and Vygen - is a good reference for matching algorithms.)
B. Korte, J. Vygen, Combinatorial Optimization,
Springer Verlag, Berlin, Heidelberg, New York, 2nd edition, 2002, 530 pages,
Very extensive book. For the German version an e-book version is available within the TU Graz network.
- A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency,
Springer, Berlin, Heidelberg, New York, 2003, 3 volumes (Volume A: 648 pages, Volume B: 570 pages, Volume C: 664 pages),
(Very comprehensive monograph on combinatorial optimization; special focus on polyhedral results and efficient algorithms, over 400 references, almost 2000 pages!)
Literature about special topics (QAP, TSP, approximation)
E. Cela, The Quadratic Assignment Problem,
Springe-Science+Business Media Dortrecht (originally published by Kluwer Adademic Publishers), 1998, 287 pages, ISBN 978-1-4419-4786-4.
Monograph on the quadratic assignment problem.
D.P. Williamson and D.B. Shmoys,
The Design of Approximation Algorithms,
Cambridge University Press, 2011, 518 pages, 978-0-521-19527-0.
Very recommendable and price-winning textbook for graduate-level courses on
approximation algorithms. Can also serve as a good reference for researchers in the area and is much more recent than the Hochbaum book.
A pdf file of the book for personal usage can be downloaded from here.
For further literature links, see the course page of my course
Combinatorial Optimization 1 and the TC course.
There will be exercise sheets made available.
Students should try to solve the exercises
on their own. The exercises should help to obtain a better understanding of
the material taught in the lecture part.
I added a number of additional exercise to point out connections to topics related to those treated in class and to encourage the students to extend their level of understanding of
combinatorial optimization and to get acquainted with problem solving on a level adequate for a course at master level.
The degree of difficulty of the problems varies. Some are very basic, others are of average
difficulty and some are actually quite challenging at this level. Feel free to as for guidance for the more challenging ones.
None yet available
Grading of the Lecture
There will be an oral exam for the lecture.
(Special arrangements possible for those who do not take part in this
year's practical.) The exams at the end of the term will be held in an online
manner (be prepared that you either need a tablet or the possibility
to film what you write on a sheet of paper).
I will not register the oral exams in TUG-Online in advance.
Please make your own appointment with me (best at least a few days before you
want to get
When preparing for the exam, aim at understanding the material presented in the lecture. It is not sufficient to learn algorithms and proofs by heart.
Grading of the Practical
Due to COVID19 there will be a specially adapted grading scheme. Details
will be discussed in the initial meeting.
Last update: March 3, 2021