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 + ε)-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 + ε)-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 + ε)-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.