CS 7250, Approximation Algorithms
Homework 2
Tue, Feb 1, 2011
Due Tue, Feb 8, 2011
Problem 1: Greedy Vertex Cover
Perhaps the first strategy one tries when designing an algorithm for an optimization problem is the
greedy strategy. For the unweighted vertex cover problem, this would involve iteratively picking
a maximum degree vertex and removing it, together with edges incident at it, until there are no
edges left.
(a) Show that this algorithm achieves an approximation guarantee of
O
(log
n
).
(b) Give a tight example: Class of input instances where this algorithm performs as bad as Ω(log
n
).
Problem 2: Set Coverage
Maximum set coverage is the following problem: Given a set
U
of
n
elements, a collection of subsets
of
U
,
S
1
, . . . , S
m
, and an integer
k
, pick
k
sets so as to maximize the number of covered elements.
(a) Show that maximum set coverage is NPhard.
(b) Show that the obvious algorithm, of greedily picking the best set in each iteration until
k
sets
are picked, achieves an approximation factor of 1

(
1

1
k
)
k
>
1

1
e
.
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 Spring '08
 Staff
 Algorithms, Graph Theory, Computational complexity theory, Approximation algorithm, following problem, Boolean satisfiability problem

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