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CS 573: Algorithms, Fall 2009
Homework 5, due Thursday, November 19, 23:59:59, 2009
Version 1.12
Name:
Net ID:
Alias:
#
Score
Grader
1.
2.
3.
4.
5.
6.
Total
Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above.
Staple this sheet to the top of your homework. If you are on campus, submit the homework by
submitting it in SC 3306 (or sliding it under the door).
In addition, the sirloin which I threw overboard, instead of drifting oﬀ into the void, didn’t seem to want to leave
the rocket and revolved about it, a second artiﬁcial satellite, which produced a brief eclipse of the sun every eleven
minutes and four seconds. To calm my nerves I calculated till evening the components of its trajectory, as well as
the orbital perturbation caused by the presence of the lost wrench. I ﬁgured out that for the next six million years
the sirloin, rotating about the ship in circular path, would lead the wrench, then catch up with it from behind and
pass it again.
– The Star Diaries, Stanislaw Lem.
Required Problems
1.
Slack form
[10 Points]
Let
L
be a linear program given in slack form, with
n
nonbasic variables
N
, and
m
basic
variables
B
. Let
N
0
and
B
0
be a diﬀerent partition of
N
∪
B
, such that

N
0
∪
B
0

=

N
∪
B

.
Show a polynomial time algorithm that computes an equivalent slack form that has
N
0
as the
nonbasic variables and
b
0
as the basic variables. How fast is your algorithm?
2.
Tedious Computations
[20 Points]
Provide
detailed
solutions for the following problems, showing each pivoting stage separately.
1
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View Full Document (A)
[5 Points]
maximize 6
x
1
+ 8
x
2
+ 5
x
3
+ 9
x
4
subject to
2
x
1
+
x
2
+
x
3
+ 3
x
4
≤
5
x
1
+ 3
x
2
+
x
3
+ 2
x
4
≤
3
x
1
,x
2
,x
3
,x
4
≥
0.
(B)
[5 Points]
maximize 2
x
1
+
x
2
subject to
2
x
1
+
x
2
≤
4
2
x
1
+ 3
x
2
≤
3
4
x
1
+
x
2
≤
5
x
1
+ 5
x
2
≤
1
x
1
,x
2
≥
0.
(C)
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This note was uploaded on 01/22/2012 for the course CS 573 taught by Professor Chekuri,c during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Chekuri,C
 Algorithms

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