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96
Chap. 15 Limits to Computation
15.8
Represent a real number in a bin as an in
f
nite column of binary digits, simi
lar to the representation of functions in Figure 15.4. Now we can use a simple
diagonalization proof. Assume that some assignment of real numbers to in
tegers is proposed. We can construct a new real number that has not been
assigned by taking the
f
rst bit of the number assigned to “1” and
F
ipping it;
take the second bit of the number assigned to “2” and
F
ip it; and so on.
15.9
Clearly, KNAPSACK is in
NP
, since we can just guess a set of items and
test in polynomial time if its size is less than
k
and its value greater than
v
.
To prove that KNAPSACK is
NP
hard, we reduce from the known
NP

complete problem EXACT KNAPSACK. EXACT KNAPSACK takes as in
put some items with sizes and a value
k
. To convert this input to an input for
KNAPSACK, we give each item a value equal to its size. We set
v
=
k
.We
now give this input to KNAPSACK.
If KNAPSACK returns “NO” then there is no solution for EXACT KNAP
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 Fall '08
 BELL,D

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