期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2011
卷号：2011
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：The bin packing problem is to find the minimum
number of bins of size one to pack a list of items with sizes
a1an in (01] . Using uniform sampling, which selects
a random element from the input list each time, we develop a
randomized O(ni=1ain(logn)(loglogn)+(1)O(1)) time (1+)-approximation
scheme for the bin packing problem. We show that every randomized
algorithm with uniform random sampling needs (nni=1ai) time to give an (1+)-approximation.
For each function s(n):NN, define (s(n)) to be
the set of all bin packing problems with the sum of item sizes equal
to s(n). For a constant b(01) , every problem in
(nb) has an O(n1−b(logn)(loglogn)+(1)O(1)) time (1+)-approximation
for an arbitrary constant . On the other hand, there is no
o(n1−b) time (1+)-approximation scheme for the bin
packing problems in (nb) for some constant 0. We
show that (nb) is NP-hard for every b(01] .
This implies a dense sublinear time hierarchy of approximation schemes for a class of NP-hard
problems, which are derived from the bin packing problem.
We also show a randomized streaming approximation scheme for the
bin packing problem such that
it needs only constant updating time and constant space,
and outputs an (1+)-approximation in (1)O(1) time.
Let S()-bin packing be the class of bin packing problems
with each input item of size at least . This research also
gives a natural example of NP-hard problem (S()-bin packing)
that has a constant time approximation scheme, and a constant time
and space sliding window streaming approximation scheme, where is a positive constant.
关键词：Approximation, bin packing, hierarchy, Sublinear time
