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 $O({n(\log\log n)\over \sum_{i=1}^n a_i}+({1\over \epsilon})^{O({1\over\epsilon})})$ time (1+e )-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+e )-approximation. For each function s (n ): N →N , define ∑(s (n )) to be the set of all bin packing problems with the sum of item sizes equal to s (n ). We show that ∑(n b ) is NP-hard for every b ∈(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+e )-approximation in $({1\over \epsilon})^{O({1\over\epsilon})}$ time. Let S (δ )-bin packing be the class of bin packing problems with each input item of size at least δ . This research also gives a natural example of NP-hard problem (S (δ )-bin packing) that has a constant time approximation scheme, and a constant time and space sliding window streaming approximation scheme, where δ is a positive constant.