**When and where: **
The class meets Tuesdays and Thursdays from 2:30PM to 4PM in room 2-131.

**Instructor:** Zarathustra Brady, room
2-350B.

**Office hours:** Wed 2:30PM-3:30PM in 2-350B.

**Prerequisites:** Linear algebra. Exposure to discrete
mathematics (18.200) is a plus, as well as exposure to algorithms
(6.006 and 18.410).

**Textbook:** There is no required textbook. Lecture
notes will be distributed during the term. For additional references,
the following textbooks are recommended (roughly in increasing difficulty
level or comprehensiveness). The last two are especially recommended
to anyone interested in a recent, in-depth coverage of the subject.

- J. Lee, A First Course in Combinatorial Optimization, Cambridge University Press, 2004.
- W. Cook, W. Cunningham, W. Pulleyblank and A. Schrijver, Combinatorial Optimization.
- C. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, 1982.
- E.L. Lawler, Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, 1976.
- G. Nemhauser and L. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, 1988.
- B. Korte and J. Vygen, Combinatorial Optimization: Theory and Algorithms, Algorithms and Combinatorics 21 Springer, Berlin Heidelberg New York, 2012. Available online with MIT certificates.
- 3-volume book by A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency , Springer-Verlag, 2003.

**Assignments and grading.** There will be roughly
bi-weekly problem sets, an in-class quiz **on Tue April 9th** and a final **on Mon May 20th from 1:30PM to 4:30PM in 56-154** (if you can't make it to class to turn in your homework, you can email it to notzeb@mit.edu instead so long as you do so before the beginning of class). There might be additional questions on psets for graduate students.
Problem sets are due in class at the beginning of the lecture.
The grade will be 30% Psets, 30% quiz and 40%
final. Attendance is strongly encouraged. Graduate students and
undergraduates may be graded differently. **Late
policy.** Late problem sets will generally not be
accepted. For a 10% discount, you can send it up to 24 hours late as a pdf attachment to the lecturer notzeb@mit.edu.

** Problem Sets:**

- Problem set 1, due 2/21/2019.
- Problem set 1 solutions.
- Problem set 2, due 3/7/2019.
- Problem set 2 solutions.
- Problem set 3, due 3/21/2019.
- Problem set 3 solutions.
- Problem set 4, due 4/9/2019.
- Problem set 4 solutions.
- Problem set 5, due 5/9/2019.
- Problem set 5 solutions.

**Syllabus:** (preliminary version)

- Introduction.
- Cardinality bipartite matching.
- Efficiency of algorithms.
- Assignment problem.
- Non-bipartite matching.
- Polytopes, linear programming, geometry.
- Polyhedral combinatorics.
- Maximum flow problem.
- Minimum cut problems.
- The ellipsoid algorithm.
- The matching polytope.
- Matroids. Matroid optimization, matroid polytope.
- Matroid intersection.
- Arborescence problem.
- Matroid union.
- Perfect graphs.

**Handouts:**

- From Lecture 1 on 2/5/2019:
- Lecture notes on bipartite matching
- Lecture notes on maximum matchings in general graphs
- Lecture notes on linear programming and polyhedral combinatorics
- Lecture notes on maximum flows and minimum cut problems
- Lecture notes on matroids
- Lecture notes on matroid intersection
- Lecture notes on the ellipsoid algorithm (For a refresher on positive semidefiniteness, see these notes on quadratic polynomials.)