FINAL EXAM STUDY GUIDE
Legend
# : you should know the statements, related facts/examples, AND PROOFS!
+ : you should know the statements and related facts/examples
- : this topic is definitely not on the final exam
Theorems
# relationships between BIBD pa

As part of its food service, a caterer needs dj napkins for each day of the upcoming week. He can buy
new napkins at the price of cents each or have his soiled napkins laundered. Two types of laundry
service are available: regular and expedited. The regul

Let A be a matrix with n columns and consider the following Linear Program,
c> x
max
(P )
such that
Ax b
Suppose that there exists an optimal solution x
of value z and let y denote the optimal solution to the dual
of (P). Consider the following Linear Pr

Consider the linear program,
maxcfw_c> x : Ax = b, x O,
(P)
where
0
5
3
c=
0 .
1
1 3 2 0 1 0
1 1 1 0
A = 0 5
0 5
6 0 2 1
1
b= 4
4
0
(a) Show that B = cfw_1, 4, 6 is a basis for (P) that is not primal feasible.
(b) Show that the reduced costs for B a

Your company MUCOW receives 3 types of dairy products from wholesales:
(1) skimmed milk which has 0% fat,
(2) high fat milk which has 3.25% fat,
(3) concentrated cream which has 50% fat,
400 litres of product (1) is available at a price of 0.8$ per litre;

Combinatorial Design Theory1
Chris Godsil
c 2010
1
version: April 9, 2010
ii
Preface
This course is an introduction to combinatorial design theory.
iii
iv
Contents
Preface
iii
1 Block Designs
1.1 Denitions . . . . . . . . . . . . . . . . . .
1.2 Examples

C&O 434/634, Winter 2011
Assignment 3
Due Friday, March 30, in class.
1. Use the multiplier theorem to construct a symmetric (37, 9, 2)-BIBD from a dierence set.
2. Let (V, B) be a design, with v points, b blocks, all of which have size k. Let Ni be the
i

C&O 434/634, Winter 2011
Assignment 2
Due Friday, March 2, in class.
1. Let Q be a km matrix whose entries are all equal to 1. Suppose that k 3 and QQt = mIk .
Prove that m is a multiple of 4.
2. Prove that the Kronecker product of two regular Hadamard ma

C&O 434/634, Winter 2011
Assignment 1
Due Monday, February 6, in class.
1. Let V be a set of size 16, whose elements are arranged into a 4 4 array. Construct a design
whose blocks are the symmetric dierence of a row and a column in the array. Show that th

(a) Let A, B, D be matrices and b, c, d be vectors of appropriate dimensions. Consider the following
Linear Programs, with variables x, z for (P1 ) and y for (P2 ),
min
c> x + d> z
such that
Ax + Bz = b
zO
(P1 )
max
c> y
such that
Ay b
Dy d
yO
(P2 )
For b