2010 | OriginalPaper | Chapter
O((logn)2) Time Online Approximation Schemes for Bin Packing and Subset Sum Problems
Authors : Liang Ding, Bin Fu, Yunhui Fu, Zaixin Lu, Zhiyu Zhao
Published in: Frontiers in Algorithmics
Publisher: Springer Berlin Heidelberg
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Given a set
S
= {
b
1
, ⋯ ,
b
n
} of integers and an integer
s
, the subset sum problem is to decide if there is a subset
S
′ of
S
such that the sum of elements in
S
′ is exactly equal to
s
. We present an online approximation scheme for this problem. It updates in
O
(log
n
) time and gives a (1 +
ε
)-approximation solution in
$O((\log n+{1\over \epsilon^2}{(\log{1\over\epsilon})^{O(1)}})\log n)$
time. The online approximation for target
s
is to find a subset of the items that have been received. The bin packing problem is to find the minimum number of bins of size one to pack a list of items
a
1
, ⋯ ,
a
n
of size in [0,1]. Let function bp(
L
) be the minimum number of bins to pack all items in the list
L
. We present an online approximate algorithm for the function bp(
L
) in the bin packing problem, where
L
is the list of the items that have been received. It updates in
O
(log
n
) updating time and gives a (1 +
ε
)-approximation solution app(
L
) for bp(
L
) in
$O((\log n)^2+({1\over \epsilon})^{O({1\over\epsilon})})$
time to satisfy app(
L
) ≤ (1 +
ε
)bp(
L
) + 1.