Homework #4
Due Date:
Wednesday, October 27th, start of class
This is a shorter homework which you are not expected to seriously work on until after the
midterm exam. However, it is recommended you attempt problem #1, which is of approximate
difficulty to what you will encounter on the midterm.
1. In the HalfCover problem, we are given
m
sets
S
1
, S
2
, ..., S
m
, each of which contains a subset
of the integers 1
,
2
, ..., n
. Our goal is to determine whether there exists a collection of
k
sets
whose union has size at least
n
2
.
(a) Suppose we prove that HalfCover is NPcomplete, and that we find an
O
(
n
4
) algorithm
for HalfCover. Does this imply that there is a polynomial algorithm for 3SAT? Does
this imply that there is an
O
(
n
4
) algorithm for 3SAT? Explain your reasoning.
(b) Prove that HalfCover is NPcomplete.
2. In the Clustering problem, we are given a weighted graph
G
= (
V, E
), an integer
k
, and a
target
T
. We want to divide the nodes into
k
sets such that any pair of nodes in the same
set have a path of length
≤
T
to each other. Prove that this problem is NPcomplete.
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 '06
 Shamsian
 Algorithms, Natural number, Prime number, Universal quantification

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