Abstract
Given a collection ℱ of subsets of
S
= {1,…,
n
},
set cover
is the problem of selecting as few as possible subsets from ℱ such that their union covers
S,
, and
max k-cover
is the problem of selecting
k
subsets from ℱ such that their union has maximum cardinality. Both these problems are NP-hard. We prove that (1 -
o
(1)) ln
n
is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low-order terms) between the ratio of approximation achievable by the greedy alogorithm (which is (1 -
o
(1)) ln n), and provious results of Lund and Yanakakis, that showed hardness of approximation within a ratio of (log
2
n
) / 2 ≃0.72 ln
n
. For max
k
-cover, we show an approximation threshold of (1 - 1/
e
)(up to low-order terms), under assumption that
P ≠ NP
.
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
Cited by
1789 articles.
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