UNIVERSITY OF MINNESOTA 
SCHOOL OF MATHEMATICS

Math 5711: Combinatorial optimization

Spring 2004

Prerequisites: Linear algebra.
Some previous exposure to graph theory may be helpful, but is definitely not necessary.
Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256
Telephone (with voice mail): 625-6682
E-mail: reiner@math.umn.edu 
Classes: Monday, Wednesday, Friday 12:20-1:10 PM
in Vincent Hall 207
Office hours: Mondays 1:25, Wednesdays 11:15, and also by appointment.  
Course content: This is a junior-senior level undergrad course on various methods and algorithms
used in combinatorial optimization. Topics we hope to discuss include:
  • Various graph optimization problems
    (e.g. minimum distances and paths, mininum cost spanning trees,
    maximum size/minimum weight matchings, network flows, stable bipartite matchings)
  • theory of linear programming
    (e.g simplex method, duality, complementary slackness)
    • and application to matrix games
  • connections with geometry, polytopes, polyehdra
  • a tiny amount of integer programming
Some topics we will likely not discuss much include
  • other methods for linear programming (e.g. dual simplex, primal-dual methods, interior point methods like ellipsoid and Karmarkar)
  • nonlinear programming, semidefinite programming
Note: Some of the same material taught in this class is also taught in Industrial Engineering 5531, and 8531 (Engineering Optimization I and II). Since this course is in mathematics, don't be surprised if the focus is a little different.
Texts: Linear programming, by Vasek Chvatal, W.H. Freeman and Co., 1983.
This should be available at the bookstore.

A course in combinatorial optimization , lecture notes by Alexander Schrijver.
Hopefully most of you can print it from here in PostScript or PDF, or maybe we'll get some copies reproduced.

Here is a handout (PostScript, PDF) on the built-in linear programming commands
in Maple, Mathematica, and MATLAB.
Other useful texts
Level Title Author(s), Publ. info Location
Same or lower Linear programming and its applications J. K. Strayer, Springer-Verlag 1989 On reserve in math library
Introduction to linear optimization D. Bertsimas and J.N. Tsitsiklis, Athena Scientific, 1997. In Walter library, call no. T57.74 .B465
Introduction to operations reserach F. Hillier and G. Lieberman, Holden-Day 1986 On reserve in math library
Linear programming: methods and applications S. Gass, McGraw-Hill 1985 On reserve in math library
Higher Theory of linear and integer programming A. Schrijver, Wiley and Sons 1998 On reserve in math library
Combinatorial optimization: algorithms and complexity C. Papadimitriou and K. Steiglitz, Dover reprints In Walter library, call no. QA402.5 .P37

I also found the following on-line list of linear programming books (by B. Engel at Purdue) to be useful.

Side topic texts
Topic Title Author(s), Publ. info Location
Graph algorithms/theorems Intro. to Graph theory, 2nd edition D. West, Prentice Hall 2001 On reserve in math library
Stable matching Stable marriage and its relation
to other combinatorial problems 		D.E. Knuth, Amer. Math. Society 1997 In math library, call no. QA164 .K5913 1997
Homework: There will be homework assignments due every other week (except weeks with exams) at the beginning of the Wednesday class, starting with Wednesday Feb. 4. The assignments will be mostly problems from either Schrijver's notes "A course in combinatorial optimization" or Chvatal's book "Linear programming". I will try to hand out brief solutions or solution outlines. Late homework will not be accepted. I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page with whom they have collaborated.   Exams and grading:  There will be two take-home midterm exams given out on dates to be determined later, each worth 15% of the course grade. There will be one take-home final exam given out either during the last week of class or final exam week, worth 20% of the course grade. The remaining 50% of the course grade will be based on the quality and quantity of homework turned in.

Both the take-home midterm and final exams are to be open-book, open-notes, but there is to be no collaboration; the only human source you will be allowed to consult is the instructor.   Policy on incompletes:  Incompletes will be given only in exceptional circumstances, where the student has completed almost the entire course with a passing grade, but something unexpected happens to prevent completion of the course. Incompletes will never be made up by taking the course again later. You must talk to me before the final exam if you think an incomplete may be warranted.   Other expectations  This is a 4-credit course, so I would guess that the average student should spend about 8 hours per week outside of class to get a decent grade. Part of this time each week would be well-spent making a first pass through the material in the book that we anticipate to cover in class that week, so that you can bring your questions/confusions to class and ask about them.
Homework assignments
Assignment or Exam Due date Problems
Homework 1 Wed Feb. 4 Schrijver's Exercises
1.1, 1.2(i), 1.4, 1.6, 1.7, 1.10
Homework 2 Wed Feb. 18 Chvatal's Exercises
1.1, 1.2, 1.3, 1.4, 1.6,
2.1(a) (via dictionaries; show each dictionary and pivot step),
2.1(b) (via tableaux; show each tableau and pivot step),
2.2, 3.1, 3.9(a,b)
Midterm exam I Wed Feb. 25 Exam I in PostScript, PDF
Homework 3 Wed 3/10 Chvatal's Exercises
5.1, 5.4, 7.1 (do 2.1(a,c) only), 9.1, 9.2, 11.1
Homework 4 Wed 3/31 Chvatal's Exercises 15.1, 15.5, 15.12
Schrijver's Exercises 2.24, 3.2, 3.3, 3.4, 3.5
Midterm exam II Wed Apr. 7 Exam II in PostScript, PDF
Homework 5 Wed 4/21 Schrijver's Exercises 3.20, 3.23(i), 5.7(i)
plus these non-Schrijver problems in PostScript, PDF
(and for a bonus challenge, try Schrijver's Exercise 3.11).
Homework 6 Fri 4/30 Schrijver's Exercises 5.8(i), 5.9, 5.11, 5.13, 5.14, 5.17
Final exam Fri 5/7 Exam I in PostScript, PDF

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