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This fle contains the exercises, hints, and solutions For Chapter 10 oF the
book ”Introduction to the Design and Analysis oF Algorithms,” 2nd edition, by
A. Levitin. The problems that might be challenging For at least some students
are marked by
±
;
those that might be diﬃcult For a majority oF students are
marked by
²
.
Exercises 10.1
1. Solve the Following linear programming problems geometrically.
(a)
maximize
3
x
+
y
subject to
−
x
+
y
≤
1
2
x
+
y
≤
4
x
≥
0
,y
≥
0
(b)
maximize
x
+2
y
subject to
4
x
≥
y
y
≤
3+
x
x
≥
0
≥
0
2. Consider the linear programming problem
minimize
c
1
x
+
c
2
y
subject to
x
+
y
≥
4
x
+3
y
≥
6
x
≥
0
≥
0
where
c
1
and
c
2
aresomerea
lnumbersnotbothequa
ltozero
.
(a) Give an example oF the coeﬃcient values
c
1
and
c
2
For which the
problem has a unique optimal solution.
(b) Give an example oF the coeﬃcient values
c
1
and
c
2
For which the
problem has infnitely many optimal solutions.
(c) Give an example oF the coeﬃcient values
c
1
and
c
2
For which the
problem does not have an optimal solution.
3. Would the solution to problem (10.2) be di±erent iF its inequality con
straints were strict, i.e.,
x
+
y<
4
and
x
6
,
respectively?
4. Trace the simplex method on
(a) the problem oF Exercise 1a.
(b) the problem oF Exercise 1b.
5. Trace the simplex method on the problem oF Example 1 in Section 6.6
1
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View Full Document(a)
±
by hand.
(b) by using one of the implementations available on the Internet.
6. Determine how many iterations the simplex method needs to solve the
problem
maximize
∑
n
j
=1
x
j
subject to
0
≤
x
j
≤
b
j
,
where
b
j
>
0
for
j
=1
,
2
,...,n.
7. Can we apply the simplex method to solve the knapsack problem (see
Example 2 in Section 6.6)?
If you answer yes, indicate whether it is a
good algorithm for the problem in question; if you answer no, explain why
not.
8.
±
Prove that no linear programming problem can have exactly
k
≥
1
optimal solutions unless
k
.
9. If a linear programming problem
maximize
∑
n
j
=1
c
j
x
j
subject to
∑
n
j
=1
a
ij
x
j
≤
b
i
for
i
,
2
,...,m
x
1
,x
2
,...,x
n
≥
0
is considered as
primal
,thenits
dual
is deFned as the linear program
ming problem
minimize
∑
m
i
=1
b
i
y
i
subject to
∑
m
i
=1
a
ij
y
i
≥
c
j
for
j
,
2
,...,n
y
1
,y
2
,...,y
m
≥
0
.
(a) Express the primal and dual problems in matrix notations.
(b) ±ind the dual of the following linear programming problem
maximize
x
1
+4
x
2
−
x
3
subject to
x
1
+
x
2
+
x
3
≤
6
x
1
−
x
2
−
2
x
3
≤
2
x
1
2
3
≥
0
.
(c) Solve the primal and dual problems and compare the optimal values
of their objective functions.
10.
±
Parliament pacifcation
In a parliament, each parliamentarian has at
most three enemies. Design an algorithm that divides the parliament into
two chambers in such a way that no parliamentarian has more than one
enemy in his or her chamber (after [Sav03], p.1, #4).
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 Spring '10
 DrT
 Algorithms

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