# 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 E-mail: reiner@math.umn.edu 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/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