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 a1, . . . , an 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) ∑n i=1 ai +( 1 2 ) 1 2 ) time (1+2)approximation scheme for the bin packing problem. We show that every randomized algorithm with uniform random sampling needs Ω( n ∑n i=1 ai ) time to give an (1 + 2)-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). For a constant b ∈ (0, 1), every problem in ∑(nb) has an O(n1−b(log n)(log log n) + ( 1 2 ) 1 2 ) time (1 + 2)-approximation for an arbitrary constant 2. On the other hand, there is no o(n1−b) time (1 + 2)-approximation scheme for the bin packing problems in ∑ (n) for some constant 2 > 0. We show that ∑ (n) 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 + 2)-approximation in ( 1 2 ) 1 2 ) 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.