Matroids - Theory and Applications - WS 2009/2010
Topics
Matroid Theory is a subject that has the power of unifying several areas and that helps to explain and discover their common properties. Studying problems in this more general setting often gives new insight into different problems and their connections.
A matroid can be basically seen as a structure that captures the essence of the notion of independence. The two best-studied fields that are generalized by the theory of matroids are linear algebra and graph theory. For example, the notion of a basis of a vectorspace and the notion of a spanning tree in graph theory as being "maximal independent sets" both generalize to the notion of a matroid basis in a corresponding matrix matroid respectively graphic matroid.
Matroids play an important role in many combinatorial optimization problems. Especially interesting are their algorithmic aspects. There is a strong connection to the Greedy Algorithm, an algorithmic concept that is able to characterize matroids, more precisely, matroids are exactly those structures where greedy strategies yield an optimal solution. As an example, think of the Kruskal Algorithm for the minimum spanning tree problem. The reason why this simple "greedy" strategy always leads to an optimal solution is the fact that the underlying considered structure is that of a matroid.
The nature of matroids of combining abstract algebraic concepts and techniques with algorithmic properties, combinatorial approaches and practical applications makes the study of them attractive in "applied" as well as in "pure" mathematics.
This lecture is an introduction to the basic ideas and concepts in matroid theory. We will consider the utilisation of matroids in combinatorial optimization and get to know the relation of matroid theory to further mathematical fields and its application to practical problems.
The following topics will be covered:
- basic terminology of matroid theory (independence system, circuit, cycle, basis, rank function, closure, etc.), matroids as generalisations of notions and concepts of linear algebra as well as graph theory
- equivalent definition and axiom systems for matroids
- matroid duality, minors, representability, matrix matroids, regular and binary matroids, graphic and cographic matroids, uniform matroids, transversal matroids, algebraic matroids
- algorithmic aspects of matroids, greedy algorithms
- application of matroid theory in combinatorial optimization, e.g. matroid polytopes and their descriptions, minimum weight (fundamental) cycle basis problems in binary, graphic and cographic matroids, complexity issues and algorithms, polyhedral aspects, formulation as integer programs, relaxations, heuristics
- applications and relation of matroid theory to further fields (coding theory, electrical engineering, computer sciene)
Prerequisites
Linear and Network Optimization, preknowledge in Integer Programming is helpful, but not required.
Homework Assignments
Nr. | Due Date | Download |
|---|---|---|
1 | November 12, 2009 | |
2 | November 19, 2009 | |
3 | November 26, 2009 | |
4 | December 3, 2009 | |
5 | December 10, 2009 | |
6 | December 17, 2009 | |
7 | January 7, 2010 | |
8 | January 14, 2010 | |
9 | January 21, 2010 | |
10 | January 28, 2010 | |
11 | February 4, 2010 | |
12 | February 11, 2010 | |
Credits
Credits for this course can be earned by an oral examination.
Students with course specialisation in "optimization" can use this lecture in the field "specialisation".
Students with course specialisation other than "optimization" can use this lecture in the field "applied mathematics" or "pure mathematics" or "general mathematics".
You can get an "Übungsschein" if you successfully work on the exercises (at least 40% of the points) and actively participate in the tutorial.
Examinations
Possible dates for the oral examination are February 24 and 25, 2010, March 8,9,10, 11, 19 and 23, 2010 (see also Prüfungsverwaltungssystem).
Please register for the exam at the office of Mrs. Eva Dengel in 14-455 (Monday-Thursday 8-12).
Literature
- J. G. Oxley: Matroid Theory, Oxford Science Publications, Oxford University Press, 1992
- D. J. A. Welsh: Matroid Theory, L.M.S. Monographs, Academic Press, 1976
Additional literature, in particular journal articles, will be presented in class.
Contact
If you have any questions, comments, suggestions, ...please tell me. (email: flobunke(at)mathematik.uni-kl.de, office: 14-443)