A Dense Hierarchy of Sublinear Time Approximation Schemes for Bin Packing

The bin packing problem is to find the minimum number of bins of size one to pack a list of items with sizes $a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n$ 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 $\mathrm{\Omega }(\frac{n}{{\sum }_{i=1}^n{a}_i})$ time to give an $(1+ϵ)$-approximation. For each function $s(n):N\to N$, 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 $ϵ$. On the other hand, there is no $o(n^{1-b})$ time $(1+ϵ)$-approximation scheme for the bin packing problems in $\sum (n^b)$ for some constant $ϵ>0$.