S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani
241
7.12. For the linear program
max
x
1

2
x
3
x
1

x
2
≤
1
2
x
2

x
3
≤
1
x
1
, x
2
, x
3
≥
0
prove that the solution
(
x
1
, x
2
, x
3
) = (3
/
2
,
1
/
2
,
0)
is optimal.
7.13.
Matching pennies.
In this simple twoplayer game, the players (call them
R
and
C
) each choose
an outcome,
heads
or
tails
. If both outcomes are equal,
C
gives a dollar to
R
; if the outcomes are
different,
R
gives a dollar to
C
.
(a) Represent the payoffs by a
2
×
2
matrix.
(b) What is the value of this game, and what are the optimal strategies for the two players?
7.14. The pizza business in Little Town is split between two rivals, Tony and Joey.
They are each
investigating strategies to steal business away from the other. Joey is considering either lowering
prices or cutting bigger slices. Tony is looking into starting up a line of gourmet pizzas, or offering
outdoor seating, or giving free sodas at lunchtime. The effects of these various strategies are
summarized in the following payoff matrix (entries are dozens of pizzas, Joey’s gain and Tony’s
loss).
T
ONY
Gourmet
Seating
Free soda
J
OEY
Lower price
+2
0

3
Bigger slices

1

2
+1
For instance, if Joey reduces prices and Tony goes with the gourmet option, then Tony will lose 2
dozen pizzas worth of business to Joey.
What is the value of this game, and what are the optimal strategies for Tony and Joey?
7.15. Find the value of the game specified by the following payoff matrix.
0
0

1

1
0
1

2

1

1

1
1
1

1
0
0
1
1

2
0

3
1

1

1

1
0

3
2

1
0

2
1

1
(
Hint:
Consider the mixed strategies
(1
/
3
,
0
,
0
,
1
/
2
,
1
/
6
,
0
,
0
,
0)
and
(2
/
3
,
0
,
0
,
1
/
3)
.)
7.16. A salad is any combination of the following ingredients: (1) tomato, (2) lettuce, (3) spinach, (4)
carrot, and (5) oil.
Each salad must contain: (A) at least 15 grams of protein, (B) at least 2
and at most 6 grams of fat, (C) at least 4 grams of carbohydrates, (D) at most 100 milligrams of
sodium. Furthermore, (E) you do not want your salad to be more than 50% greens by mass. The
nutritional contents of these ingredients (per 100 grams) are
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242
Algorithms
ingredient
energy
protein
fat
carbohydrate
sodium
(kcal)
(grams)
(grams)
(grams)
(milligrams)
tomato
21
0.85
0.33
4.64
9.00
lettuce
16
1.62
0.20
2.37
8.00
spinach
371
12.78
1.58
74.69
7.00
carrot
346
8.39
1.39
80.70
508.20
oil
884
0.00
100.00
0.00
0.00
Find a linear programming applet on the Web and use it to make the salad with the fewest
calories under the nutritional constraints. Describe your linear programming formulation and
the optimal solution (the quantity of each ingredient and the value). Cite the Web resources that
you used.
7.17. Consider the following network (the numbers are edge capacities).
A
B
C
D
T
S
7
6
3
4
2
2
5
9
(a) Find the maximum flow
f
and a minimum cut.
(b) Draw the residual graph
G
f
(along with its edge capacities). In this residual network, mark
the vertices reachable from
S
and the vertices from which
T
is reachable.
(c) An edge of a network is called a
bottleneck edge
if increasing its capacity results in an
increase in the maximum flow. List all bottleneck edges in the above network.
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 Spring '11
 Linear Programming, Algorithms, Graph Theory, Computational complexity theory, NPcomplete, U.V. Vazirani

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