The bin packing problem is to find the minimum
number of bins of size one to pack a list of items with sizes
$a_1,\ldots , a_n$ in $(0,1]$. Using uniform sampling, which selects
a random element from the input list each time, we develop a
randomized $O({n(\log n)(\log\log n)\over \sum_{i=1}^n a_i}+({1\over
\epsilon})^{O({1\over\epsilon})})$ time $(1+\epsilon)$-approximation
scheme for the bin packing problem. We show that every randomized
algorithm with uniform random sampling needs $\Omega({n\over
\sum_{i=1}^n a_i})$ time to give an $(1+\epsilon)$-approximation.
For each function $s(n): N\rightarrow N$, define $\sum(s(n))$ to be
the set of all bin packing problems with the sum of item sizes equal
to $s(n)$. For a constant $b\in (0,1)$, every problem in
$\sum(n^{b})$ has an $O(n^{1-b}(\log n)(\log\log n)+({1\over
\epsilon})^{O({1\over\epsilon})})$ time $(1+\epsilon)$-approximation
for an arbitrary constant $\epsilon$. On the other hand, there is no
$o(n^{1-b})$ time $(1+\epsilon)$-approximation scheme for the bin
packing problems in $\sum(n^{b})$ for some constant $\epsilon>0$. We
show that $\sum(n^{b})$ is NP-hard for every $b\in (0,1]$.
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 $(1+\epsilon)$-approximation in $({1\over
\epsilon})^{O({1\over\epsilon})}$ time.
Let $S(\delta)$-bin packing be the class of bin packing problems
with each input item of size at least $\delta$. This research also
gives a natural example of NP-hard problem ($S(\delta)$-bin packing)
that has a constant time approximation scheme, and a constant time
and space sliding window streaming approximation scheme, where $\delta$ is a positive constant.