TR11-028 Authors: Richard Beigel, Bin Fu

Publication: 4th March 2011 15:06

Downloads: 2005

Keywords:

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.