Introduction to Algorithms
Solution Set 9
CS 482, Spring 2008
(1)
Let
T
be the number of trucks used by the algorithm, and suppose the trucks are labeled
1
,
2
, . . . , T
in the order that the algorithm loads them up. Observe that for
i
= 1
,
2
, . . . , T

1,
the combined weight in trucks
i
and
i
+ 1 must be at least
K
+ 1; otherwise the first item that
was loaded into truck
i
+ 1 would actually have fit in truck
i
, violating the specification of the
algorithm.
We group the trucks into
P
=
d
T/
2
e
pairs: trucks 1 and 2, trucks 3 and 4, etc. By our previous
observation, each of the first
P

1 pairs of trucks contains a combined weight of at least
K
+ 1.
So the combined weight of all the items is at least (
P

1)(
K
+ 1)
.
If
W
denotes the combined
weight of all the items and
OPT
denotes the minimum possible number of trucks, then
OPT
≥
W/K
≥
(
P

1)
K
+ 1
K
> P.
Hence
OPT
≥
P
≥
T/
2, which proves that the algorithm is a 2approximation.
(2)
(a).
Set Cover
. Given a universal set
U
and a collection of subsets
S
1
, S
2
, . . . , S
m
⊆ U
, what
is the minimum size of a subcollection
{
S
i
1
, S
i
2
, . . . , S
i
k
}
whose union is
U
?
Answer:
min
m
X
i
=1
x
i
s.t.
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 Spring '08
 KLEINBERG
 Algorithms, Set Theory, Graph Theory, Optimization, combined weight, ye ye xu, constraint ye xu

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