Combinatorial Optimization
Problem set 6
Assigned Monday, June 22, 2015. Due Thursday, June 25, 2015.
1. Suppose a simple undirected graph has more than one minimum spanning tree. Can Prims
algorithm (or Kruskals algorithm) be used to nd all of them? Expl

Combinatorial Optimization
Problem set 8
Assigned Thursday, July 2, 2015. Due Thursday, July 9, 2015.
1. Fix constants a R and b > 1. For n N, let f (n) = na and g(n) = bn . Prove that
f (n) = o g(n) .
2. Carefully state the decision (recognition) version

Combinatorial Optimization
Problem set 4
Assigned Tuesday, June 9, 2015. Due Friday, June 12, 2015.
1. Consider a project consisting of the following nine activities.
Activity
Immediate
prerequisites
Usual time
(days)
Crash time
(days)
Cost per day
to spe

Combinatorial Optimization
Problem set 2: solutions
1. Consider the following two linear programs in standard form:
maximize
subject to
cT x
cT x
maximize
Ax = b
subject to
Ax = b
x0
x0
Can both of these linear programs have feasible solutions with arbitr

Combinatorial Optimization
Final examination
Tuesday, July 14, 2015
Please read all of the following before beginning this exam.
1. This exam consists of six problems. Each of the problems is worth 20 points.
2. You may use the textbook, your course notes

Combinatorial Optimization
Problem set 3: solutions
1. In class (and on the Analysis of a simplex tableau handout), I claimed that if a simplex
tableau (for a non-degenerate linear program) contains a column having a negative entry
in the objective row an

Combinatorial Optimization
Problem set 6: solutions
1. Suppose a simple undirected graph has more than one minimum spanning tree. Can Prims
algorithm (or Kruskals algorithm) be used to nd all of them? Explain why or why not,
and give an example.
Solution.

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Combinatorial Optimization
Problem set 5
Assigned Monday, June 15, 2015. Due Thursday, June 18, 2015.
1. Consider the problem of determining the least expensive way to complete a project by
a given deadline (as we studied last week). When the linear progr

Combinatorial Optimization
Problem set 3
Assigned Thursday, June 4, 2015. Due Monday, June 8, 2015.
For your convenience, a tool to help with pivoting simplex tableaux is available online at
http:/www.math.cmu.edu/~bkell/pivot.html.
1. In class (and on th

Combinatorial Optimization
Problem set 1
Assigned Thursday, May 28, 2015. Due Monday, June 1, 2015.
1. Formulate a linear program for the following optimization problem. Then solve your
linear program with Maple and interpret the results. (Please include

Combinatorial Optimization
Problem set 4: solutions
1. Consider a project consisting of the following nine activities.
Activity
Immediate
prerequisites
Usual time
(days)
Crash time
(days)
Cost per day
to speed up
A
B
C
D
E
F
G
H
I
A
B
B
C, D
D
F
E, F, G
2

Combinatorial Optimization
Problem set 1: solutions
1. Formulate a linear program for the following optimization problem. Then solve your linear
program with Maple and interpret the results.
The Ace Rening Company produces two types of unleaded gasoline,

Combinatorial Optimization
Problem set 2
Assigned Monday, June 1, 2015. Due Thursday, June 4, 2015.
1. Consider the following two linear programs in standard form:
maximize
subject to
cT x
maximize
Ax = b
subject to
x0
cT x
Ax = b
x0
Can both of these lin

Combinatorial Optimization
Problem set 5: solutions
1. Consider the problem of determining the least expensive way to complete a project by a
given deadline. When the linear program is formulated, the objective function has a constant
term. For instance,

Combinatorial Optimization
Problem set 7
Assigned Friday, June 26, 2015. Due Wednesday, July 1, 2015.
1. Formulate and solve an integer program for the following scenario.
A trader of unusual objects is traveling with a caravan that begins in city A,
proc

Combinatorial Optimization
Problem set 8: solutions
1. Fix constants a R and b > 1. For n N, let f (n) = na and g(n) = bn . Prove that
f (n) = o g(n) .
Solution. First we observe that g(n) 0 for all n N and limn g(n) = limn bn =
because b > 1. Now we con

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IP formulation examples
June 24, 2015
1. Machine shop reopening. A small machine shop is reopening after a re had forced it
to close for extensive repairs. The shop has three product lines: plates, gears, and housings.
Each product line requires specializ