Math 5711: Combinatorial optimization

Spring 2006

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): (612) 625-6682
Classes: Monday, Wednesday 11:55 A.M. - 01:10 P.M. in Vincent Hall 211
Office hours: Mon, Tue, Wed at 11:15am, also Wed at 3pm, and 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:
  • theory of linear programming (mostly from Chvatal's text)
    (e.g simplex method, duality, complementary slackness) and application to matrix games
  • network/graph optimization problems (mostly from Schrijver's text)
    (e.g. minimum distances and paths, mininum cost spanning trees,
    maximum size/minimum weight matchings, network flows, stable bipartite matchings)
  • 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,
    and interior point methods like ellipsoid and Karmarkar's algorithm
  • 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,
e.g. if we prove things more, and ask you to explain or prove things more often.
Texts: There are two texts we will use, and our homework will come from both:
  • 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.
    You can access or print them from here in PostScript or PDF.
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 Optimization in operations research Ronald L. Rardin, Prentice Hall 1998 In Walter library, call no.T57.7 .R37 1998
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 research 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
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 and exams: There will be 6 homework assignments due generally every other week, except for
  • the 6th homework will be only one week (see the schedule below),
  • 2 weeks where there will be a week-long take-home midterm exam,
  • a week at the end with a week-long take-home final exam.
Tentative dates for the assignments and exams are in the schedule below. The take-home midterms and final exam are open-book, open-library, open-web, but no collaboration or consultation of human sources is allowed.

Late homework will not be accepted. Early homework is fine, and can be left in my mailbox in the School of Math mailroom near Vincent Hall 105. Collaboration is encouraged as long as everyone collaborating understands thoroughly the solution, and you write up the solution in your own words, along with a note at the top of the homework indicating with whom you've collaborated.

Homework solutions should be well-explained-- the grader will be told not to give credit for an unsupported answer.

Grading:  Homework = 50% of grade
Each of 2 midterms = 15% of grade
Final exam 20% of grade.

Complaints about the grading should be brought to me.

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/exam schedule and assignments (tentative)
Assignment or Exam Due date Problems from the text
Homework 1 Wed Feb. 1 All from Chvatal:
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)
Homework 2 Wed Feb. 15 All from Chvatal:
4.1(a,b), 17.1(a,b), 5.1(a,b), 5.4, (9.1, 9.2 removed from this HW)
Midterm exam 1 Wed. Feb. 22 Midterm exam 1 in PostScript, PDF
Homework 3 Wed Mar. 8 All from Chvatal:
7.1(a), 9.3, 11.1, 15.1, 15.5, 15.12
Homework 4 Wed Mar. 29 From Schrijver:
2.24, 1.1, 1.2(i), 1.4, 1.6 (1.7 removed from this HW)
Not from Schrijver:
Solve the integer programming problem max{y: 8x+3y <= 12, -3x+y <= 0, x,y >=0, integers}
(a) graphically (i.e. by inspection of a picture),
(b) using branch-and-bound, in which each LP is solved graphically,
(c) using Gomory cutting planes (with each LP solved by any means, including computer)
Midterm exam 2 Wed. Apr. 5 Midterm exam 2 in PostScript, PDF
Homework 5 Wed Apr. 19 From Schrijver:
1.7, 1.10, 3.2, 3.3, 3.4, 3.5, 3.17, 3.18, 3.23(i)
Homework 6 Wed Apr. 26 (note 1-week due date!) From Schrijver: 5.7(i)
Not from Schrijver:
Problems 2 and 3 regarding the Gale-Shapley algorithm
from this extra problem sheet in PostScript, PDF
Final exam Wed. May 3 Final exam in PostScript, PDF

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