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 $a_1,..., a_n$ in $(0,1]$. Using uniform sampling, which selects a random element from the input list each time, we develop a randomized $O({n(\log n)(\log\log n)\over \sum_{i=1}^n a_i}+({1\over ε})^{O({1\overε})})$ time $(1+ε)$-approximation scheme for the bin packing problem. We show that every randomized algorithm with uniform random sampling needs $Ω({n\over \sum_{i=1}^n a_i})$ time to give an $(1+ε)$-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 $O(n^{1-b}(\log n)(\log\log n)+({1\over ε})^{O({1\overε})})$ time $(1+ε)$-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 $ε>0$.

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